Circuit Implementation and Antisynchronization of an Improved Lorenz Chaotic System
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Li Xiong
2016-01-01
Full Text Available An improved Lorenz chaotic system is proposed, making it into a circuit which is easy to be implemented by using some basic electronic components. The antisynchronization error systems can be asymptotically stabilized at the origin with three different methods which are proposed to control the improved Lorenz system. Theoretical analyses and simulation results are given to demonstrate the feasibility and effectiveness of these proposed schemes. Then the hardware circuit for the proposed Lorenz system is implemented by repeated optimization design. Experimental results show that the circuit has good comprehensive performance.
Chaotic attractor transforming control of hybrid Lorenz-Chen system
Institute of Scientific and Technical Information of China (English)
Qi Dong-Lian; Wang Qiao; Gu Hong
2008-01-01
Based on passive theory, this paper studies a hybrid chaotic dynamical system from the mathematics perspective to implement the control of system stabilization.According to the Jacobian matrix of the nonlinear system, the stabilization control region is gotten.The controller is designed to stabilize fast the minimum phase Lorenz-Chen chaotic system after equivalently transforming from chaotic system to passive system. The simulation results show that the system not only can be controlled at the different equilibria, but also can be transformed between the different chaotic attractors.
Hybrid TS fuzzy modelling and simulation for chaotic Lorenz system
Institute of Scientific and Technical Information of China (English)
Li De-Quan
2006-01-01
The projection of the chaotic attractor observed from the Lorenz system in the X-Z plane is like a butterfly, hence the classical Lorenz system is widely known as the butterfly attractor, and has served as a prototype model for studying chaotic behaviour since it was coined. In this work we take one step further to investigate some fundamental dynamic behaviours of a novel hybrid Takagi-Sugeno (TS) fuzzy Lorenz-type system, which is essentially derived from the delta-operator-based TS fuzzy modelling for complex nonlinear systems, and contains the original Lorenz system of continuous-time TS fuzzy form as a special case. By simply and appropriately tuning the additional parametric perturbations in the two-rule hybrid TS fuzzy Lorenz-type system, complex (two-wing) butterfly attractors observed from this system in the three dimensional (3D) X-Y-Z space are created, which have not yet been reported in the literature, and the forming mechanism of the compound structures have been numerically investigated.
Robust chaotic control of Lorenz system by backstepping design
Energy Technology Data Exchange (ETDEWEB)
Peng, C.-C. [Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan (China); Chen, C.-L. [Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan (China)], E-mail: chiehli@mail.ncku.edu.tw
2008-07-15
This work presents a robust chaotic control strategy for the Lorenz chaos via backstepping design. Backstepping technique is a systematic tool of control law design to provide Lyapunov stability. The concept of extended system is used such that a continuous sliding mode control (SMC) effort is generated using backstepping scheme. In the proposed control algorithm, an adaptation law is applied to estimate the system parameter and the SMC offers the robustness to model uncertainties and external disturbances so that the asymptotical convergence of tracking error can be achieved. Regarding the SMC, an equivalent control algorithm is chosen based on the selection of Lyapunov stability criterion during backstepping approach. The converging rate of error state is relative to the corresponding dynamics of sliding surface. Numerical simulations demonstrate its advantages to a regulation problem and an orbit tracking problem of the Lorenz chaos.
Adaptive fuzzy sliding mode control of Lorenz chaotic system
Institute of Scientific and Technical Information of China (English)
WU Ligang; WANG Changhong
2007-01-01
By using the exponential reaching law technology,a sliding mode controller was designed for Lorenz chaotic system subject to an unknown external disturbance.On this basis,considering the unknown disturbance,an adaptive law was introduced to adaptively estimate the parameters of the disturbance bounds.Furthermore,to eliminate the chattering resulting from the discontinuous switch controller and to guarantee system transient performance,a new adaptive fuzzy sliding mode controller was designed.The results of the simulation show the effectiveness of the proposed control scheme.
Directory of Open Access Journals (Sweden)
Ping Zhou
2012-01-01
Full Text Available The unstable equilibrium points of the fractional-order Lorenz chaotic system can be controlled via fractional-order derivative, and chaos synchronization for the fractional-order Lorenz chaotic system can be achieved via fractional-order derivative. The control and synchronization technique, based on stability theory of fractional-order systems, is simple and theoretically rigorous. The numerical simulations demonstrate the validity and feasibility of the proposed method.
Ping Zhou; Rui Ding
2012-01-01
The unstable equilibrium points of the fractional-order Lorenz chaotic system can be controlled via fractional-order derivative, and chaos synchronization for the fractional-order Lorenz chaotic system can be achieved via fractional-order derivative. The control and synchronization technique, based on stability theory of fractional-order systems, is simple and theoretically rigorous. The numerical simulations demonstrate the validity and feasibility of the proposed method.
Directory of Open Access Journals (Sweden)
Cheng-Hsiung Yang
2013-01-01
Full Text Available A new symplectic chaos synchronization of chaotic systems with uncertain chaotic parameters is studied. The traditional chaos synchronizations are special cases of the symplectic chaos synchronization. A sufficient condition is given for the asymptotical stability of the null solution of error dynamics and a parameter difference. The symplectic chaos synchronization with uncertain chaotic parameters may be applied to the design of secure communication systems. Finally, numerical results are studied for symplectic chaos synchronized from two identical Lorenz-Stenflo systems in three different cases.
Parameter estimation of Lorenz chaotic system using a hybrid swarm intelligence algorithm
Energy Technology Data Exchange (ETDEWEB)
Lazzús, Juan A., E-mail: jlazzus@dfuls.cl; Rivera, Marco; López-Caraballo, Carlos H.
2016-03-11
A novel hybrid swarm intelligence algorithm for chaotic system parameter estimation is present. For this purpose, the parameters estimation on Lorenz systems is formulated as a multidimensional problem, and a hybrid approach based on particle swarm optimization with ant colony optimization (PSO–ACO) is implemented to solve this problem. Firstly, the performance of the proposed PSO–ACO algorithm is tested on a set of three representative benchmark functions, and the impact of the parameter settings on PSO–ACO efficiency is studied. Secondly, the parameter estimation is converted into an optimization problem on a three-dimensional Lorenz system. Numerical simulations on Lorenz model and comparisons with results obtained by other algorithms showed that PSO–ACO is a very powerful tool for parameter estimation with high accuracy and low deviations. - Highlights: • PSO–ACO combined particle swarm optimization with ant colony optimization. • This study is the first research of PSO–ACO to estimate parameters of chaotic systems. • PSO–ACO algorithm can identify the parameters of the three-dimensional Lorenz system with low deviations. • PSO–ACO is a very powerful tool for the parameter estimation on other chaotic system.
Parameter estimation of Lorenz chaotic system using a hybrid swarm intelligence algorithm
Lazzús, Juan A.; Rivera, Marco; López-Caraballo, Carlos H.
2016-03-01
A novel hybrid swarm intelligence algorithm for chaotic system parameter estimation is present. For this purpose, the parameters estimation on Lorenz systems is formulated as a multidimensional problem, and a hybrid approach based on particle swarm optimization with ant colony optimization (PSO-ACO) is implemented to solve this problem. Firstly, the performance of the proposed PSO-ACO algorithm is tested on a set of three representative benchmark functions, and the impact of the parameter settings on PSO-ACO efficiency is studied. Secondly, the parameter estimation is converted into an optimization problem on a three-dimensional Lorenz system. Numerical simulations on Lorenz model and comparisons with results obtained by other algorithms showed that PSO-ACO is a very powerful tool for parameter estimation with high accuracy and low deviations.
Institute of Scientific and Technical Information of China (English)
Cang Shi-Jian; Chen Zeng-Qiang; Wu Wen-Juan
2009-01-01
This paper presents a non-autonomous hyper-chaotic system, which is formed by adding a periodic driving signal to a four-dimensional chaotic model extended from the Lorenz system. The resulting non-autonomous hyper-chaotic system can display any dynamic behaviour among the periodic orbits, intermittency, chaos and hyper-chaos by controlling the frequency of the periodic signal. The phenomenon has been well demonstrated by numerical simulations, bifurcation analysis and electronic circuit realization. Moreover, the system is concrete evidence for the presence of Pomeau-Manneville Type-Ⅰ intermittency and crisis-induced intermittency. The emergence of a different type of intermittency is similarly subjected to the frequency of periodic forcing. By statistical analysis, power scaling laws consisting in different intermittency are obtained for the lifetime in the laminar state between burst states.
Dynamics of the stochastic Lorenz chaotic system with long memory effects
Energy Technology Data Exchange (ETDEWEB)
Zeng, Caibin, E-mail: zeng.cb@mail.scut.edu.cn; Yang, Qigui, E-mail: qgyang@scut.edu.cn [School of Mathematics, South China University of Technology, Guangzhou 510640 (China)
2015-12-15
Little seems to be known about the ergodic dynamics of stochastic systems with fractional noise. This paper is devoted to discern such long time dynamics through the stochastic Lorenz chaotic system (SLCS) with long memory effects. By a truncation technique, the SLCS is proved to generate a continuous stochastic dynamical system Λ. Based on the Krylov-Bogoliubov criterion, the required Lyapunov function is further established to ensure the existence of the invariant measure of Λ. Meanwhile, the uniqueness of the invariant measure of Λ is proved by examining the strong Feller property, together with an irreducibility argument. Therefore, the SLCS has exactly one adapted stationary solution.
Dynamics of the stochastic Lorenz chaotic system with long memory effects
Zeng, Caibin; Yang, Qigui
2015-12-01
Little seems to be known about the ergodic dynamics of stochastic systems with fractional noise. This paper is devoted to discern such long time dynamics through the stochastic Lorenz chaotic system (SLCS) with long memory effects. By a truncation technique, the SLCS is proved to generate a continuous stochastic dynamical system Λ. Based on the Krylov-Bogoliubov criterion, the required Lyapunov function is further established to ensure the existence of the invariant measure of Λ. Meanwhile, the uniqueness of the invariant measure of Λ is proved by examining the strong Feller property, together with an irreducibility argument. Therefore, the SLCS has exactly one adapted stationary solution.
Adaptive set-point tracking of the Lorenz chaotic system using non-linear feedback
Energy Technology Data Exchange (ETDEWEB)
Haghighatdar, F. [Department of Electronic Engineering, University of Isfahan, Hezar-Jerib St., Postal code: 8174673441, Isfahan (Iran, Islamic Republic of)], E-mail: fr_haghighat@yahoo.com; Ataei, M. [Department of Electronic Engineering, University of Isfahan, Hezar-Jerib St., Postal code: 8174673441, Isfahan (Iran, Islamic Republic of)], E-mail: mataei1971@yahoo.com
2009-05-30
In this paper, an adaptive control method for set-point tracking of the Lorenz chaotic system by using non-linear feedback is proposed. The design procedure of the proposed controller is accomplished in two steps. At the first step, using Lyapunov's direct method, a non-linear state feedback is selected so that without any need to apply identification techniques, in despite of the uncertain parameters existence in the system state equations, the asymptotic stability of the general Lorenz system is guaranteed in a stochastic point of the manifold containing general system equilibrium points. At the second step, a linear state feedback with adaptive gain is added to the prior controller to eliminate the tracking error. In order to guarantee the system asymptotic stability at desired set-point, the indirect Lyapunov's method is used. Finally, to show the effectiveness of the proposed methodology, the simulation results of different experiments including system parameters changes and set-point variation are provided.
Piecewise affine models of chaotic attractors: The Rössler and Lorenz systems
Amaral, Gleison F. V.; Letellier, Christophe; Aguirre, Luis Antonio
2006-03-01
This paper proposes a procedure by which it is possible to synthesize Rössler [Phys. Lett. A 57, 397-398 (1976)] and Lorenz [J. Atmos. Sci. 20, 130-141 (1963)] dynamics by means of only two affine linear systems and an abrupt switching law. Comparison of different (valid) switching laws suggests that parameters of such a law behave as codimension one bifurcation parameters that can be changed to produce various dynamical regimes equivalent to those observed with the original systems. Topological analysis is used to characterize the resulting attractors and to compare them with the original attractors. The paper provides guidelines that are helpful to synthesize other chaotic dynamics by means of switching affine linear systems.
Directory of Open Access Journals (Sweden)
Tiezheng Song
2012-01-01
Full Text Available Signals with weak singularities are important for condition monitoring, fault forecasting, and medicine diagnosis. However, the weak singularity in a signal is usually hidden by strong noise. A novel fast method is proposed for detecting a weak singularity in a noised signal by determining a critical threshold towards chaos for the Lorenz system. First, a rough critical threshold value is calculated by local Lyapunov exponents with a step size 0.1. Second, the exact threshold value is calculated by the bisection algorithm. The advantage of the method will not only reduce the computation costs, but also show the weak singular signal which can be accurately identified from strong noise. When the variance of an external signal method embeds into a Lorenz system, according to the parametric equivalent relation between the Lorenz system and the original system, the critical threshold value of the parameter in a Lorenz system is determined.
Bounds for the chaotic region in the Lorenz model
Barrio, Roberto; Serrano, Sergio
2009-08-01
In a previous paper, the authors made an extensive numerical study of the Lorenz model, changing all three parameters of the system. We conjectured that the region of parameters where the Lorenz model is chaotic is bounded for fixed r. In this paper, we give a theoretical proof of the conjecture by obtaining theoretical bounds for the chaotic region and by using Fenichel theory. The theoretical bounds are complemented with numerical studies performed using the Maximum Lyapunov Exponent and OFLI2 techniques, and a comparison of both sets of results is shown. Finally, we provide a complete three-dimensional model of the chaotic regime depending on the three parameters.
A radial basis function sliding mode controller for chaotic Lorenz system
Energy Technology Data Exchange (ETDEWEB)
Guo Huijun [School of Electrical Engineering, Xi' an Jiaotong University, Xi' an 710049 (China)]. E-mail: realghj@yahoo.com.cn; Lin Suifang [Department of Automation, Xi' an University of Technology, Xi' an 710048 (China); Liu Junhua [School of Electrical Engineering, Xi' an Jiaotong University, Xi' an 710049 (China)
2006-03-06
This Letter presents a novel method to controlling Lorenz chaos via an adaptive radial basis function sliding mode controller. The proposed scheme combines the advantages of the adaptive control, neural network and sliding mode control strategies without precise system model information. It has on-line learning ability to deal with the parametric uncertainty and disturbance by adjusting the control parameters. A sliding mode controller is designed via the Lyapunov stability theory in order to guarantee the high quality of the controlled system. The simulation results show that this method is feasible and effective for chaos control, and the robustness to parametric changes and extern disturbance is provided.
Predictability of Forced Lorenz Systems
Li, Baosheng; Ding, Ruiqiang; Li, Jianping; Zhong, Quanjia
2017-04-01
Based on the nonlinear local Lyapunov exponent (NLLE) approach, the influences of external forcing on the predictability are studied in the Lorenz systems with constant and quasi-periodic forces in this paper. The results indicate that for the Lorenz systems with constant and quasi-periodic forces, their predictability limits increase with the forcing strength. With the same magnitude and different directions, the constant or quasi-periodic forcing shows different effects on the predictability limit in the Lorenz system, and these effects become significant with the increase of the forcing strength. Generally speaking, the positive forcing leads to a higher predictability limit than the negative forcing. Therefore, when we think about the effects of positive and negative elements and phases in the atmosphere and ocean research, the predictability problems driven by different phases should be considered separately. In addition, the influences of constant and quasi-periodic forces on the predictability are different in the Lorenz system. The effect of the constant forcing on the predictability is mainly reflected in the linear phase of error growth, while the nonlinear phase should also be considered for the situation of the quasi-periodic forcing. The predictability limit of the system under constant forcing is longer than the system under quasi-periodic forcing. These results based on simple chaotic model could provide insight into the studies of the actual atmosphere predictability.
Institute of Scientific and Technical Information of China (English)
LIAO Xiaoxin; FU Yuli; XIE Shengli
2005-01-01
Constructing a family of generalized Lyapunov functions, a new method is proposed to obtain new global attractive set and positive invariant set of the Lorenz chaotic system. The method we proposed greatly simplifies the complex proofs of the two famous estimations presented by the Russian scholar Leonov. Our uniform formula can derive a series of the new estimations. Employing the idea of intersection in set theory, we extract a new Leonov formula-like estimation from the family of the estimations. With our method and the new estimation, one can confirm that there are no equilibrium, periodic solutions, almost periodic motions, wandering motions or other chaotic attractors outside the global attractive set. The Lorenz butterfly-like singular attractors are located in the global attractive set only. This result is applied to the chaos control and chaos synchronization. Some feedback control laws are obtained to guarantee that all the trajectories of the Lorenz systems track a periodic solution, or globally stabilize an unstable (or locally stable but not globally asymptotically stable) equilibrium. Further, some new global exponential chaos synchronization results are presented. Our new method and the new results are expected to be applied in real secure communication systems.
Global stabilization of a Lorenz system
Institute of Scientific and Technical Information of China (English)
李世华; 田玉平
2003-01-01
In this paper,using feedback linearizing technique,we show that a Lorenz system can be considered as a cascade system.Moreover,this system satisfies the assumptions of global stabilization of cascade systems.Thus continuous state feedback control laws are proposed to globally stabilize the Lorenz system at the equilibrium point.Simulation results are presented to verify our method.This method can be further generalized to other chaotic systems such as Chen system,coupled dynamos system,etc.
Li, Chunhe; Wang, Erkang; Wang, Jin
2012-05-21
We developed a potential flux landscape theory to investigate the dynamics and the global stability of a chemical Lorenz chaotic strange attractor under intrinsic fluctuations. Landscape was uncovered to have a butterfly shape. For chaotic systems, both landscape and probabilistic flux are crucial to the dynamics of chaotic oscillations. Landscape attracts the system down to the chaotic attractor, while flux drives the coherent motions along the chaotic attractors. Barrier heights from the landscape topography provide a quantitative measure for the robustness of chaotic attractor. We also found that the entropy production rate and phase coherence increase as the molecular numbers increase. Power spectrum analysis of autocorrelation function provides another way to quantify the global stability of chaotic attractor. We further found that limit cycle requires more flux and energy to sustain than the chaotic strange attractor. Finally, by detailed analysis we found that the curl probabilistic flux may provide the origin of the chaotic attractor.
Stability Analysis and Design of Impulsive Control Lorenz Systems Family
Institute of Scientific and Technical Information of China (English)
YU Yong-Bin; ZHANG Hong-Bin; ZHANG Feng-Li; YU Jue-Bang; LIAO Xiao-Feng
2009-01-01
Lorenz systems family unifying Lorenz system, Chen system and Lu system is a typical chaotic family.In this paper, we consider impulsive control Lorenz chaotic systems family with time-varying impulse intervals. By establishing an effective tool of a set of inequalities, we analyze the asymptotic stability of impulsive control Lorenz systems family and obtain some new less conservative conditions. Based on the stability analysis, we design a novel impulsive controller with time-varying impulse intervals. Illustrative examples are provided to show the feasibility and effectiveness of our method. The obtained results not only can be used to design impulsive control for Lorenz systems family, but also can be extended to other chaotic systems.
An Anti-Control Scheme for Spiral under Lorenz Chaotic Signals
Institute of Scientific and Technical Information of China (English)
MA Jun; YING He-Ping; PU Zhong-Sheng
2005-01-01
@@ The Fitzhugh-Nagumo (FHN) equation is used to generate spiral and spatiotemporal chaos. The weak Lorenz chaotic signalis imposed on the system locally and globally. It is found that for the right chaotic driving signal,spiral and spatiotemporal chaos can be suppressed. The simulation results also show that this anti-control scheme is effective so that the system emerges into the stable states quickly after a short duration of chaotic driving (about 50 time units) and the continuous driving keeps the system in a homogeneous state.
Chaotic gas turbine subject to augmented Lorenz equations.
Cho, Kenichiro; Miyano, Takaya; Toriyama, Toshiyuki
2012-09-01
Inspired by the chaotic waterwheel invented by Malkus and Howard about 40 years ago, we have developed a gas turbine that randomly switches the sense of rotation between clockwise and counterclockwise. The nondimensionalized expressions for the equations of motion of our turbine are represented as a starlike network of many Lorenz subsystems sharing the angular velocity of the turbine rotor as the central node, referred to as augmented Lorenz equations. We show qualitative similarities between the statistical properties of the angular velocity of the turbine rotor and the velocity field of large-scale wind in turbulent Rayleigh-Bénard convection reported by Sreenivasan et al. [Phys. Rev. E 65, 056306 (2002)]. Our equations of motion achieve the random reversal of the turbine rotor through the stochastic resonance of the angular velocity in a double-well potential and the force applied by rapidly oscillating fields. These results suggest that the augmented Lorenz model is applicable as a dynamical model for the random reversal of turbulent large-scale wind through cessation.
Dynamics of the Lorenz Robbins system with control
Huang, Debin; Zhang, Lizhen
2006-06-01
In this paper, the existence of periodic orbits and homoclinic orbits in the Lorenz equations with high Rayleigh number r, i.e., the Lorenz-Robbins system, is rigorously proved by the generalized Melnikov method for the three-dimensional slowly varying systems. We analyze stability of these periodic orbits and show that the existence of these nontransverse but symmetrical homoclinic orbits implies the existence of chaos in the Lorenz-Robbins system. The results obtained analytically show the existence of chaotic dynamics in the Lorenz-Robbins system for the first time, but also solve a disagreement on the conditions of existence of periodic orbits in the system. In addition, a simple adaptive algorithm, which was recently developed by the author [D. Huang, Stabilizing near-nonhyperbolic chaotic systems with applications, Phys. Rev. Lett. 93 (2004) 214101] for stabilizing the near-nonhyperbolic chaotic systems, is used to successfully control the chaotic mixing of the Lorenz flows with high Rayleigh number found.
Directory of Open Access Journals (Sweden)
Rodrigo Méndez-Ramírez
2017-01-01
Full Text Available This paper presents a new three-dimensional autonomous chaotic system. The proposed system generates a chaotic attractor with the variation of two parameters. Analytical and numerical studies of the dynamic properties to generate chaos, for continuous version (CV and discretized version (DV, for the new chaotic system (NCS were conducted. The CV of the NCS was implemented by using an electronic circuit with operational amplifiers (OAs. In addition, the presence of chaos for DV of the NCS was proved by using the analytical and numerical degradation tests; the time series was calculated to determine the behavior of Lyapunov exponents (LEs. Finally, the DV of NCS was implemented, in real-time, by using a novel embedded system (ES Mikromedia Plus for PIC32MX7 that includes one microcontroller PIC32 and one thin film transistor touch-screen display (TFTTSD, together with external digital-to-analog converters (DACs.
Estimating the ultimate bound and positively invariant set for a generalized Lorenz system
Institute of Scientific and Technical Information of China (English)
SHU Yong-lu; ZHANG Yong-hao
2008-01-01
A generalized Lyapunov function was employed to investigate the ultimate bound and positively invariant set of a generalized Lorenz system. We derived an ellipsoidal estimate of the ultimate bound and positively invariant set for the generalized Lorenz system, for all the positive values of system parameters a, b, and c. Our results extend the related result of Li, et al. [Li DM, Lu JA, Wu XQ, et al., Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system, Journal of Mathematical Analysis and Application, 2006, 323(2): 844-653].
Chaos control of Lorenz systems using adaptive controller with input saturation
Energy Technology Data Exchange (ETDEWEB)
Yau, H.-T. [Department of Electrical Engineering, Far-East College, No 49 Jung-Haw Road, Hsin-Shih Town, Tainan 744, Taiwan (China)]. E-mail: pan1012@ms52.hinet.net; Chen, C.-L. [Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan 701, Taiwan (China)]. E-mail: chiehli@mail.ncku.edu.tw
2007-12-15
This paper presents an adaptive sliding mode control scheme for Lorenz chaos subject to saturating input. The state of Lorenz system can be asymptotically driven to an equilibrium point in spite of the presence of input saturation and external disturbance using the proposed control scheme. Numerical simulations demonstrate the effectiveness of its application to chaotic system control. It also shows that the settling time will be decreased, if the saturation bound of control input is relaxed.
Experimental Confirmation of a Modified Lorenz System
Institute of Scientific and Technical Information of China (English)
LIU Ling; LIU Chong-Xin; ZHANG Yan-Bin
2007-01-01
We experimentally demonstrate the butterfly-shaped chaotic attractor we have proposed before [Int. J. Nonlin.Sci. Numerical Simulation 7 (2006) 187]. Some basic dynamical properties and chaotic behaviour of this new butterfly attractor are studied and they are in agreement with the results of our theoretical analysis. Moreover,the proposed system is experimental demonstrated.
Institute of Scientific and Technical Information of China (English)
祝大伟; 涂俐兰
2013-01-01
In this paper, Lorenz chaotic system with stochastic perturbation and unknown parameters is investigated, in which the stochastic perturbations is one-dimensional random process of the standard Wiener. Based on stochastic Lyapunov stability theory, Itˆo formula and adaptive control method combined with three adaptive control laws and two adaptive control laws respectively, two mean square Asymptotic adaptive synchronization standards are put forward theoretically. These new standards are in a simple form and easy to deal with. Moreover, with these standards, not only drive system with stochastic perturbations can be synchronized with the respond system, but also unknown parameters in the system can be identified. Finally, the Matlab numerical simulations confirm that the proposed results are correct and effective.% 本论文研究了具有随机扰动和未知参数的Lorenz混沌系统，其中随机扰动是一维标准Wiener随机过程。基于随机李雅普洛夫稳定性理论、Itˆo (伊藤)公式以及自适应控制方法，本文分别通过设置三个和两个控制器，从理论上提出了两个均方渐近自适应同步标准，这些标准简单易行，不仅能使得随机扰动下的驱动系统和响应系统达到均方渐近同步，而且能同时识别出系统中的未知参数。最后的Matlab数值模拟验证了提出的理论结果的正确性和有效性。
Institute of Scientific and Technical Information of China (English)
宋鑫超; 苏庆堂; 赵永升
2016-01-01
为解决当前图像加密算法采用独立的置乱与扩散操作，降低算法内联性，且忽略混沌序列生成存在的时延因素，使其难以抵御明文攻击等不足，提出一种内联时延混沌映射耦合 Lorenz系统的图像加密算法。将时间延迟引入 Logistic 映射中，生成 Arnold映射的初值；基于明文像素点，构造 Arnold映射迭代次数计算模型；根据 Arnold映射的迭代次数，建立其映射控制参数的计算函数，生成一组随机序列，利用位置集合，完成图像置乱；迭代超混沌 Lorenz 系统，生成4D序列组；引入密钥流，修正4D序列组；构造像素扩散机制，完成图像加密。实验结果表明，与当前加密结构相比，该算法拥有更高的保密性能与更强的密钥敏感性。%Using current image encryption algorithm is difficult to resist plaintext attacks induced by independently scrambling and diffusion operating resulting in reducing the algorithm’s inline,also the existing time delay factor of the chaos sequence is ig-nored,so the image encryption algorithm based on inline time delay chaotic map coupled with Lorenz system was proposed.The initial value of Arnold map was generated by introducing the time delay into the Logistic map.The iteration number model of Ar-nold map was constructed based on the pixels of plaintext.The initial value of Arnold map was generated by introducing the time delay into the Logistic map and basing on iteration number.The calculation function of the mapping control parameters was es-tablished based on the initial value for iterating to produce the random sequence and the image was permutated by position set. The 4D sequence group was generated by iterating the hyper-chaotic system.The pixel diffusion mechanism was constructed by the modified 4D sequence group with key stream to finish image encryption.Experimental results show that this algorithm has better security performance and stronger key
Increased-order generalized synchronization of chaotic and hyperchaotic systems
Indian Academy of Sciences (India)
K S Ojo; S T Ogunjo; A N Njah; I A Fuwape
2015-01-01
This paper presents increased-order generalized synchronization (GS) of chaotic and hyperchaotic systems with different order based on active control technique. By this technique, we design suitable control functions to achieve GS between (i) a new three-dimensional (3D) chaotic system and four-dimensional (4D) hyperchaotic Lorenz system and (ii) four-dimensional hyperchaotic Lorenz system and five-dimensional (5D) hyperchaotic Lorenz system. The corresponding numerical simulation results are presented to verify the effectiveness of this technique.
Lag Synchronization of Complex Lorenz System with Applications to Communication
Directory of Open Access Journals (Sweden)
Fangfang Zhang
2015-07-01
Full Text Available In communication, the signal at the receiver end at time t is the signal from the transmitter side at time t −Τ (Τ ≥ 0 and it is the lag time as the time lag of transmission. Therefore, lag synchronization (LS is more accurate than complete synchronization to design communication scheme. Taking complex Lorenz system as an example, we design the LS controller according to error feedback. Using chaotic masking, we propose a communication scheme based on LS and independent component analysis (ICA. It is suitable to transmit multiple messages with all kinds of amplitudes and it has the ability of anti-noise. Numerical simulations verify the feasibility and effectiveness of the presented schemes.
Dynamical analysis of fractional-order Rössler and modified Lorenz systems
Energy Technology Data Exchange (ETDEWEB)
Letellier, Christophe, E-mail: Christophe.Letellier@coria.fr [Université de Rouen – CORIA, BP 12, F-76801 Saint-Etienne du Rouvray Cedex (France); Aguirre, Luis A. [Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, 31270-901 Belo Horizonte, MG (Brazil)
2013-10-15
This Letter is devoted to the dynamical analysis of fractional-order systems, namely the Rössler and a modified Lorenz system. The work here described compares the dynamical regimes of such fractional-order systems to that of the corresponding standard systems. It turns out that most of the chaotic attractors are topologically equivalent to those found in the original integer-order systems, although in some particular (and apparently rare) cases unusual bifurcation patterns and attractors are found.
Dynamical analysis of fractional-order Rössler and modified Lorenz systems
Letellier, Christophe; Aguirre, Luis A.
2013-10-01
This Letter is devoted to the dynamical analysis of fractional-order systems, namely the Rössler and a modified Lorenz system. The work here described compares the dynamical regimes of such fractional-order systems to that of the corresponding standard systems. It turns out that most of the chaotic attractors are topologically equivalent to those found in the original integer-order systems, although in some particular (and apparently rare) cases unusual bifurcation patterns and attractors are found.
Control uncertain continuous-time chaotic dynamical system
Institute of Scientific and Technical Information of China (English)
齐冬莲; 赵光宙
2003-01-01
The new chaos control method presented in this paper is useful for taking advantage of chaos. Based on sliding mode control theory, this paper provides a switching manifold controlling strategy of chaotic system, and also gives a kind of adaptive parameters estimated method to estimate the unknown systems' parameters by which chaotic dynamical system can be synchronized. Taking the Lorenz system as example, and with the help of this controlling strategy, we can synchronize chaotic systems with unknown parameters and different initial conditions.
Control uncertain continuous-time chaotic dynamical system.
Qi, Dong-Lian; Zhao, Guang-Zhou
2003-01-01
The new chaos control method presented in this paper is useful for taking advantage of chaos. Based on sliding mode control theory, this paper provides a switching manifold controlling strategy of chaotic system, and also gives a kind of adaptive parameters estimated method to estimate the unknown systems' parameters by which chaotic dynamical system can be synchronized. Taking the Lorenz system as example, and with the help of this controlling strategy, we can synchronize chaotic systems with unknown parameters and different initial conditions.
Localization of Compact Invariant Sets of the Lorenz'1984 System
Directory of Open Access Journals (Sweden)
Kh. M. Ramazanova
2015-01-01
Full Text Available Localization of compact invariant sets of a dynamical system is one way to conduct a qualitative analysis of dynamical system. The localization task is aimed at evaluating the location of invariant compact sets of systems, which are equilibrium, periodic trajectories, attractors and repellers, and invariant tori. Such sets and their properties largely determine the structure of the phase portrait of the system. For this purpose, one can use a localization set, i.e. a set in the phase space of the system that contains all invariant compact sets.This article considers the problem of localization of invariant compact sets of an Autonomous version of the Lorenz-84 system. The system represents a simple model of the General circulation of the atmosphere in middle latitudes. The model was used in various climatological studies. To build localization set of the system the so-called functional localization method is applied. The article describes the main provisions of this method, lists the main properties of the localization sets. The simplest version of the Lorenz-84 system when there are no thermal loads is analyzed, and a common variant of the Autonomous Lorenz-84 system, in which for some values of system parameters chaotic dynamics occurs is investigated. In the first case it is shown that the only invariant compact set of the system is its equilibrium position, and localization function turned out to be a Lyapunov function of the system. For the General version of the system a family of localization sets is built and the intersection of this family is described. Graphical illustration for the localization set at fixed values of the parameters is shown. The result of the study partially overlaps with the result of K.E. Starkov on the subject, but provides additional information.The theme of localization of invariant compact sets is discussed quite actively in the literature. Research focuses both on the development of the method and its
Limit Cycles near Stationary Points in the Lorenz System
Institute of Scientific and Technical Information of China (English)
YANG Shi-Pu; ZHU Ke-Qin; ZHOU Xiao-Zhou
2005-01-01
@@ The limit cycles in the Lorenz system near the stationary points are analysed numerically. A plane in phase space of the linear Lorenz system is used to locate suitable initial points of trajectories near the limit cycles. The numerical results show a stable and an unstable limit cycle near the stationary point. The stable limit cycle is smaller than the unstable one and has not been previously reported in the literature. In addition, all the limit cycles in the Lorenz system are theoretically proven not to be planar.
Tracking control of chaotic dynamical systems with feedback linearization
Institute of Scientific and Technical Information of China (English)
QI Dong-lian; MA Guo-jin
2005-01-01
A new method was proposed for tracking the desired output of chaotic dynamical system using the feedback linearization and nonlinear extended statement observer method. The feedback linearization was used to convert the nonlinear chaotic system into linear system. The extended Luenberger-like statements observer was designed to reconstructing and observing the unmeasured statements when the tracking controller was designed. By this way, the chaotic system could be forced to track variable desired output, which could be a time variant function or an equilibrium points.Taken the Lorenz chaotic system as example, the simulation results show the validity of the conclusion and effectiveness of the algorithm.
Institute of Scientific and Technical Information of China (English)
张芳芳; 刘树堂; 余卫勇
2013-01-01
Self-synchronization of time delay implies that the synchronization between the time-delay system and the original system keeps the structure and parameters of systems unchanged, thus these various problems produced by time-delay in practice are avoided. Taking a time-delay complex Lorenz system for example, we investigate its dynamic characteristics and the influence of of time lag factor. A nonlinear feedback controller is designed to realize the self-synchronization of time delay of the complex Lorenz system. Numerical simulations verify the effectiveness of the presented controller. The controller adopts some states to realize the synchronization of all states. It is simple in principle and easy to implement in engineering.%自时滞同步是指保持混沌系统结构和参数不变的情况下，使时滞系统和原系统同步，从而避免了现实中因为时滞而产生的各种问题。本文以时滞复Lorenz系统为例，研究其动态特性及时滞因数的影响，并提出了一种非线性反馈控制器实现了复Lorenz系统的自时滞混沌同步。数值仿真结果验证了该控制器的有效性。该控制器只对部分状态进行控制，实现了所有状态的同步，原理简单，易于工程实现。
Complex dynamical behavior and chaos control in fractional-order Lorenz-like systems
Institute of Scientific and Technical Information of China (English)
Li Rui-Hong; Chen Wei-Sheng
2013-01-01
In this paper,the complex dynamical behavior of a fractional-order Lorenz-like system with two quadratic terms is investigated.The existence and uniqueness of solutions for this system are proved,and the stabilities of the equilibrium points are analyzed as one of the system parameters changes.The pitchfork bifurcation is discussed for the first time,and the necessary conditions for the commensurate and incommensurate fractional-order systems to remain in chaos are derived.The largest Lyapunov exponents and phase portraits are given to check the existence of chaos.Finally,the sliding mode control law is provided to make the states of the Lorenz-like system asymptotically stable.Numerical simulation results show that the presented approach can effectively guide chaotic trajectories to the unstable equilibrium points.
Function projective synchronization of different chaotic systems with uncertain parameters
Energy Technology Data Exchange (ETDEWEB)
Du Hongyue [Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin 150001 (China)], E-mail: du_hong_yue@yahoo.com.cn; Zeng Qingshuang; Wang Changhong [Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin 150001 (China)
2008-08-11
This Letter investigates the function projective synchronization of different chaotic systems with unknown parameters. By Lyapunov stability theory, the adaptive control law and the parameter update law are derived to make the states of two different chaotic systems asymptotically synchronized up to a desired scaling function. Numerical simulations on Lorenz system and Newton-Leipnik system are presented to verify the effectiveness of the proposed scheme.
Chaos control of chaotic dynamical systems using backstepping design
Energy Technology Data Exchange (ETDEWEB)
Yassen, M.T. [Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516 (Egypt)] e-mail: mtyassen@yahoo.com
2006-01-01
This work presents chaos control of chaotic dynamical systems by using backstepping design method. This technique is applied to achieve chaos control for each of the dynamical systems Lorenz, Chen and Lue systems. Based on Lyapunov stability theory, control laws are derived. We used the same technique to enable stabilization of chaotic motion to a steady state as well as tracking of any desired trajectory to be achieved in a systematic way. Numerical simulations are shown to verify the results.
Passivity-Based Synchronization of Unified Chaotic System
Directory of Open Access Journals (Sweden)
K. Kemih
2008-01-01
Full Text Available This letter further improves and extends the work of Kemih et al. In detail, feedback passivity synchronization with only one controller for a unified chaotic system is discussed here. It is noticed that the unified system contains the noted Lorenz, Lu, and Chen systems. Numerical simulations are given to show the effectiveness of these methods.
Anti-synchronization Between Lorenz and Liu Hyperchaotic Systems
Institute of Scientific and Technical Information of China (English)
ZHENG Qiang; ZHANG Xiao-Ping; REN Zhong-Zhou
2008-01-01
Anti-synchronization between different hyperchaotic systems is presented using Lorenz and Liu systems.When the parameters of two systems are known,one can use active synchronization.When the parameters are unknown or uncertain,the adaptive synchronization is applied.The simulation results verify the effectiveness of the proposed two schemes for anti-synchronization between different hyperehaotic systems.
Synchronization of chaotic systems.
Pecora, Louis M; Carroll, Thomas L
2015-09-01
We review some of the history and early work in the area of synchronization in chaotic systems. We start with our own discovery of the phenomenon, but go on to establish the historical timeline of this topic back to the earliest known paper. The topic of synchronization of chaotic systems has always been intriguing, since chaotic systems are known to resist synchronization because of their positive Lyapunov exponents. The convergence of the two systems to identical trajectories is a surprise. We show how people originally thought about this process and how the concept of synchronization changed over the years to a more geometric view using synchronization manifolds. We also show that building synchronizing systems leads naturally to engineering more complex systems whose constituents are chaotic, but which can be tuned to output various chaotic signals. We finally end up at a topic that is still in very active exploration today and that is synchronization of dynamical systems in networks of oscillators.
Synchronization of chaotic systems
Energy Technology Data Exchange (ETDEWEB)
Pecora, Louis M.; Carroll, Thomas L. [U.S. Naval Research Laboratory, Washington, District of Columbia 20375 (United States)
2015-09-15
We review some of the history and early work in the area of synchronization in chaotic systems. We start with our own discovery of the phenomenon, but go on to establish the historical timeline of this topic back to the earliest known paper. The topic of synchronization of chaotic systems has always been intriguing, since chaotic systems are known to resist synchronization because of their positive Lyapunov exponents. The convergence of the two systems to identical trajectories is a surprise. We show how people originally thought about this process and how the concept of synchronization changed over the years to a more geometric view using synchronization manifolds. We also show that building synchronizing systems leads naturally to engineering more complex systems whose constituents are chaotic, but which can be tuned to output various chaotic signals. We finally end up at a topic that is still in very active exploration today and that is synchronization of dynamical systems in networks of oscillators.
Stochastic impulsive control for the stabilization of Lorenz system
Institute of Scientific and Technical Information of China (English)
Wang Liang; Zhao Rui; Xu Wei; Zhang Ying
2011-01-01
This paper derives some sufficient conditions for the stabilization of Lorenz system with stochastic impulsive control. The estimate of the upper bound of impulse interval for asymptotically stable control is obtained. Some differences between the system with stochastic impulsive control and with deterministic impulsive control are presented. Computer simulation is given to show the effectiveness of the proposed method.
Frustrated synchronization in competing drive-response coupled chaotic systems
Sinha, S
1998-01-01
Chaotic systems can be synchronized by linking them to a common signal, subject to certain conditions. However, the presence of multiple driving signals coming from different systems, give rise to novel behavior. The particular case of Lorenz systems, with two independent systems driving another system through drive-response coupling has been studied in this paper. This is the simplest arrangement which shows the effect of ``frustrated synchronization'' due to competition between the two driver systems. The resulting response system attractor deviates significantly from the conventional Lorenz attractor. A new measure of desynchronization is proposed, which shows a power-law scaling with the competition parameter.
Chaos control and synchronization for a special generalized Lorenz canonical system - The SM system
Energy Technology Data Exchange (ETDEWEB)
Liao Xiaoxin [Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei 430074 (China); Xu, F. [Department of Applied Mathematics, University of Western Ontario, London, Ontario, N6A 5B7 (Canada); Wang, P. [School of Automation, Wuhan University of Technology, Wuhan, Hubei 430070 (China); Yu Pei [Department of Applied Mathematics, The University of Western Ontario, London, Ontario, N6A 5B7 (Canada)], E-mail: pyu@pyu1.apmaths.uwo.ca
2009-03-15
This paper presents some simple feedback control laws to study global stabilization and global synchronization for a special chaotic system described in the generalized Lorenz canonical form (GLCF) when {tau} = -1 (which, for convenience, we call Shimizu-Morioka system, or simply SM system). For an arbitrarily given equilibrium point, a simple feedback controller is designed to globally, exponentially stabilize the system, and reach globally exponent synchronization for two such systems. Based on the system's coefficients and the structure of the system, simple feedback control laws and corresponding Lyapunov functions are constructed. Because all conditions are obtained explicitly in terms of algebraic expressions, they are easy to be implemented and applied to real problems. Numerical simulation results are presented to verify the theoretical predictions.
Adaptive Control of the Chaotic System via Singular System Approach
Directory of Open Access Journals (Sweden)
Yudong Li
2014-01-01
Full Text Available This paper deals with the control problem of the chaotic system subject to disturbance. The sliding mode surface is designed by singular system approach, and sufficient condition for convergence is given. Then, the adaptive sliding mode controller is designed to make the state arrive at the sliding mode surface in finite time. Finally, Lorenz system is considered as an example to show the effectiveness of the proposed method.
Chen, Diyi; Zhang, Runfan; Sprott, J C; Chen, Haitao; Ma, Xiaoyi
2012-06-01
In this paper, we focus on the synchronization between integer-order chaotic systems and a class of fractional-order chaotic system using the stability theory of fractional-order systems. A new sliding mode method is proposed to accomplish this end for different initial conditions and number of dimensions. More importantly, the vector controller is one-dimensional less than the system. Furthermore, three examples are presented to illustrate the effectiveness of the proposed scheme, which are the synchronization between a fractional-order Chen chaotic system and an integer-order T chaotic system, the synchronization between a fractional-order hyperchaotic system based on Chen's system and an integer-order hyperchaotic system, and the synchronization between a fractional-order hyperchaotic system based on Chen's system and an integer-order Lorenz chaotic system. Finally, numerical results are presented and are in agreement with theoretical analysis.
Sepantaie, Marc M.; Namazi, Nader M.; Sepantaie, Amir M.
2016-05-01
This paper is devoted to addressing the synchronization, and detection of random binary data exposed to inherent channel variations existing in Free Space Optical (FSO) communication systems. This task is achieved by utilizing the identical synchronization methodology of Lorenz chaotic communication system, and its synergetic interaction in adversities imposed by the FSO channel. Moreover, the Lorenz system has been analyzed, and revealed to induce Stochastic Resonance (SR) once exposed to Additive White Gaussian Noise (AWGN). In particular, the resiliency of the Lorenz chaotic system, in light of channel adversities, has been attributed to the success of the proposed communication system. Furthermore, this paper advocates the use of Haar wavelet transform for enhanced detection capability of the proposed chaotic communication system, which utilizes Chaotic Parameter Modulation (CPM) technique for means of transmission.
Adaptive control of chaotic systems based on a single layer neural network
Energy Technology Data Exchange (ETDEWEB)
Shen Liqun [Space Control and Inertia Technology Research Center, Harbin Institute of Technology, Harbin 150001 (China)], E-mail: liqunshen@gmail.com; Wang Mao [Space Control and Inertia Technology Research Center, Harbin Institute of Technology, Harbin 150001 (China)
2007-08-27
This Letter presents an adaptive neural network control method for the chaos control problem. Based on a single layer neural network, the dynamic about the unstable fixed period point of the chaotic system can be adaptively identified without detailed information about the chaotic system. And the controlled chaotic system can be stabilized on the unstable fixed period orbit. Simulation results of Henon map and Lorenz system verify the effectiveness of the proposed control method.
The equal combination synchronization of a class of chaotic systems with discontinuous output
Energy Technology Data Exchange (ETDEWEB)
Luo, Runzi; Zeng, Yanhui [Department of Mathematics, Nanchang University, Nanchang 330031 (China)
2015-11-15
This paper investigates the equal combination synchronization of a class of chaotic systems. The chaotic systems are assumed that only the output state variable is available and the output may be discontinuous state variable. By constructing proper observers, some novel criteria for the equal combination synchronization are proposed. The Lorenz chaotic system is taken as an example to demonstrate the efficiency of the proposed approach.
Liu, Ping
2013-07-01
This paper deals with the finite-time stabilization of unified chaotic complex systems with known and unknown parameters. Based on the finite-time stability theory, nonlinear control laws are presented to achieve finite-time chaos control of the determined and uncertain unified chaotic complex systems, respectively. The two controllers are simple, and one of the uncertain unified chaotic complex systems is robust. For the design of a finite-time controller on uncertain unified chaotic complex systems, only some of the unknown parameters need to be bounded. Simulation results for the chaotic complex Lorenz, Lü and Chen systems are presented to validate the design and analysis.
Comparison between different synchronization methods of identical chaotic systems
Energy Technology Data Exchange (ETDEWEB)
Haeri, Mohammad [Advanced Control System Laboratory, Electrical Engineering Department, Sharif University of Technology, Azadi Avenue, P.O. Box 11365-9363 Tehran (Iran, Islamic Republic of)]. E-mail: haeri@sina.sharif.edu; Khademian, Behzad [Advanced Control System Laboratory, Electrical Engineering Department, Sharif University of Technology, Azadi Avenue, P.O. Box 11365-9363 Tehran (Iran, Islamic Republic of)
2006-08-15
This paper studies and compares three nonadaptive (bidirectional, unidirectional, and sliding mode) and two adaptive (active control and backstepping) synchronization methods on the synchronizing of four pairs of identical chaotic systems (Chua's circuit, Roessler system, Lorenz system, and Lue system). Results from computer simulations are presented in order to illustrate the effectiveness of the methods and to compare them based on different criteria.
Cascade Chaotic System With Applications.
Zhou, Yicong; Hua, Zhongyun; Pun, Chi-Man; Chen, C L Philip
2015-09-01
Chaotic maps are widely used in different applications. Motivated by the cascade structure in electronic circuits, this paper introduces a general chaotic framework called the cascade chaotic system (CCS). Using two 1-D chaotic maps as seed maps, CCS is able to generate a huge number of new chaotic maps. Examples and evaluations show the CCS's robustness. Compared with corresponding seed maps, newly generated chaotic maps are more unpredictable and have better chaotic performance, more parameters, and complex chaotic properties. To investigate applications of CCS, we introduce a pseudo-random number generator (PRNG) and a data encryption system using a chaotic map generated by CCS. Simulation and analysis demonstrate that the proposed PRNG has high quality of randomness and that the data encryption system is able to protect different types of data with a high-security level.
Two routes to chaos in the fractional Lorenz system with dimension continuously varying
Peng, Guojun; Jiang, Yaolin
2010-10-01
The object of this paper is to reveal the relation between dynamics of the fractional system and its dimension defined as a sum of the orders of all involved derivatives. We take the fractional Lorenz system as example and regard one or three of its orders as bifurcation parameters. In this framework, we compute the corresponding bifurcation diagrams via an optimal Poincaré section technique developed by us and find there exist two routes to chaos when its dimension increases from some values to 3. One is the process of cascaded period-doubling bifurcations and the other is a crisis (boundary crisis) which occurs in the evolution of chaotic transient behavior. We would like to point out that our investigation is the first to find out that a fractional differential equations (FDEs) system can evolve into chaos by the crisis. Furthermore, we observe rich dynamical phenomena in these processes, such as two-stage cascaded period-doubling bifurcations, chaotic transients, and the transition from coexistence of three attractors to mono-existence of a chaotic attractor. These are new and interesting findings for FDEs systems which, to our knowledge, have not been described before.
Adaptive Control on a Class of Uncertain Chaotic Systems
Institute of Scientific and Technical Information of China (English)
LIU Guo-Gang; ZHAO Yi
2005-01-01
@@ By using a simple combination of feedback entrainment control (with an updating feedback strength) and adaptive scheme, for a large class of chaotic systems, it is proven rigorously by using the invariance principle of differential equations that all unknown model parameters can be estimated dynamically and the uncertain system can be controlled to an arbitrary desired smooth orbit. The illustration of the Lorenz system and the corresponding numerical results on the effect of noise are given.
LORENZ: a system for planning long-bone fracture reduction
Birkfellner, Wolfgang; Burgstaller, Wolfgang; Wirth, Joachim; Baumann, Bernard; Jacob, Augustinus L.; Bieri, Kurt; Traud, Stefan; Strub, Michael; Regazzoni, Pietro; Messmer, Peter
2003-05-01
Long bone fractures belong to the most common injuries encountered in clinical routine trauma surgery. Preoperative assessment and decision making is usually based on standard 2D radiographs of the injured limb. Taking into account that a 3D - imaging modality such as computed tomography (CT) is not used for diagnosis in clinical routine, we have designed LORENZ, a fracture reduction planning tool based on such standard radiographs. Taking into account the considerable success of so-called image free navigation systems for total knee replacement in orthopaedic surgery, we assume that a similar tool for long bone fracture reposition should have considerable impact on computer-aided trauma surgery in a standard clinical routine setup. The case for long bone fracture reduction is, however, somewhat more complicated since not only scale independent angles indicating biomechanical measures such as varus and valgus are involved. Reduction path planning requires that the individual anatomy and the classification of the fracture is taken into account. In this paper, we present the basic ideas of this planning tool, it's current state, and the methodology chosen. LORENZ takes one or more conventional radiographs of the broken limb as input data. In addition, one or more x-rays of the opposite healthy bone are taken and mirrored if necessary. A most adequate CT model is being selected from a database; currently, this is achieved by using a scale space approach on the digitized x-ray images and comparing standard perspective renderings to these x-rays. After finding a CT-volume with a similar bone, a triangulated surface model is generated, and the surgeon can break the bone and arrange the fragments in 3D according to the x-ray images of the broken bone. Common osteosynthesis plates and implants can be loaded from CAD-datasets and are visualized as well. In addition, LORENZ renders virtual x-ray views of the fracture reduction process. The hybrid surface/voxel rendering
Control uncertain continuous－time chaotic dynamical system
Institute of Scientific and Technical Information of China (English)
齐冬莲; 赵光宙
2003-01-01
The new chaos control method presented in this paper is useful for taking advantage of chaos.Based on sliding mode control theory, this paper provides a switching manifold controlling strategy of chaoticsystem, and also gives a kind of adaptive parameters estimated method to estimate the unknown systems' pa-rameters by which chaotic dynamical system can be synchronized. Taking the Lorenz system as example, and with the help of this controlling strategy, we can synchronize chaotic systems with unknown parameters and different initial conditions.
Open-loop frequency response for a chaotic masking system
Institute of Scientific and Technical Information of China (English)
Huang Xian-Gao; Yu Pei; Huang Wei
2006-01-01
In this paper, a new numerical simulation approach is proposed for the study of open-loop frequency response of a chaotic masking system. Using Chua's circuit and the Lorenz system as illustrative examples, we have shown that one can employ chaos synchronization to separate the feedback network from a chaotic masking system, and then use numerical simulation to obtain the open-loop synchronization response, the phase response, and the amplitude response of a chaotic masking system. Based on the analysis of the frequency response, we have also proved that changing the amplitude of the exciting (input) signal within normal working domain does not influence the frequency response of the chaotic masking system. The new numerical simulation method developed in this paper can be extended to consider the open-loop frequency response of other systems described by differential or difference equations.
Compound Generalized Function Projective Synchronization for Fractional-Order Chaotic Systems
Directory of Open Access Journals (Sweden)
Chunde Yang
2016-01-01
Full Text Available A modified function projective synchronization for fractional-order chaotic system, called compound generalized function projective synchronization (CGFPS, is proposed theoretically in this paper. There are one scaling-drive system, more than one base-drive system, and one response system in the scheme of CGFPS, and the scaling function matrices come from multidrive systems. The proposed CGFPS technique is based on the stability theory of fractional-order system. Moreover, we achieve the CGFPS between three-driver chaotic systems, that is, the fractional-order Arneodo chaotic system, the fractional-order Chen chaotic system, and the fractional-order Lu chaotic system, and one response chaotic system, that is, the fractional-order Lorenz chaotic system. Numerical experiments are demonstrated to verify the effectiveness of the CGFPS scheme.
Chaos control of 4D chaotic systems using recursive backstepping nonlinear controller
Energy Technology Data Exchange (ETDEWEB)
Laoye, J.A. [Nonlinear and Statistical Physics Research Group, Department of Physics, Olabisi Onabanjo University, P.M.B. 2002, Ago-Iwoye (Nigeria); Vincent, U.E. [Nonlinear and Statistical Physics Research Group, Department of Physics, Olabisi Onabanjo University, P.M.B. 2002, Ago-Iwoye (Nigeria)], E-mail: ue_vincent@yahoo.com; Kareem, S.O. [Nonlinear and Statistical Physics Research Group, Department of Physics, Olabisi Onabanjo University, P.M.B. 2002, Ago-Iwoye (Nigeria)
2009-01-15
This paper examines chaos control of two four-dimensional chaotic systems, namely: the Lorenz-Stenflo (LS) system that models low-frequency short-wavelength gravity waves and a new four-dimensional chaotic system (Qi systems), containing three cross products. The control analysis is based on recursive backstepping design technique and it is shown to be effective for the 4D systems considered. Numerical simulations are also presented.
Synchronization of N different coupled chaotic systems with ring and chain connections
Institute of Scientific and Technical Information of China (English)
LIU Yan; L(U) Ling
2008-01-01
Synchronization of N different coupled chaotic systems with ring and chain Lorenz system, and the R(o)ssler system are used as examples in verifying effectiveness of the method. Based on the Lyapunov stability theory, the form of the controller is designed and the area of the coupling coefficients is determined. Simulations indicate that global synchronization of the N different chaotic systems can be realized by choosing appropriate coupling coefficients by using the controller.
a New Color Image Encryption Based on High-Dimensional Chaotic Systems
Li, Pi; Wang, Xing-Yuan; Fu, Hong-Jing; Xu, Da-Hai; Wang, Xiu-Kun
2014-12-01
The high-dimensional chaotic systems (HDCS) have a lot of advantages as more multifarious mechanism, greater the key space, more ruleless for the time series of the system variable than with the low-dimensional chaotic systems (LDCS), etc. Thus, a novel encryption scheme using Lorenz system is suggested. Moreover, we use substitution-diffusion architecture to advance the security of the scheme. The theoretical and experimental results show that the suggested cryptosystem has higher security.
Synchronization of noise-perturbed generalized Lorenz system by sliding mode control*
Institute of Scientific and Technical Information of China (English)
Kong Cui-Cui; Chen Shi-Hua
2009-01-01
Synchronization of a noise-perturbed generalized Lorenz system by using sliding mode control method is investigated in this paper. Two sliding mode control methods are proposed to synchronize the noise-perturbed generalized Lorenz system. Numerical simulations are also provided for the illustration and verification of the methods.
Tracking control and synchronization of chaotic systems based upon sampled-data feedback
Institute of Scientific and Technical Information of China (English)
陈士华; 刘杰; 谢进; 陆君安
2002-01-01
A novel tracking control and synchronization method is proposed based upon sampled-data feedback. This methodcan make a chaotic system approach any desired smooth orbit and synchronize the driving system and the responsesystem, both in the same structure and in diverse structures. Finally, a numerical simulation with a Lorenz system isprovided for the purpose of illustration and verification.
Altmann, Eduardo G; Tél, Tamás
2013-01-01
There are numerous physical situations in which a hole or leak is introduced in an otherwise closed chaotic system. The leak can have a natural origin, it can mimic measurement devices, and it can also be used to reveal dynamical properties of the closed system. In this paper we provide an unified treatment of leaking systems and we review applications to different physical problems, both in the classical and quantum pictures. Our treatment is based on the transient chaos theory of open systems, which is essential because real leaks have finite size and therefore estimations based on the closed system differ essentially from observations. The field of applications reviewed is very broad, ranging from planetary astronomy and hydrodynamical flows, to plasma physics and quantum fidelity. The theory is expanded and adapted to the case of partial leaks (partial absorption/transmission) with applications to room acoustics and optical microcavities in mind. Simulations in the lima .con family of billiards illustrate...
Chaotic systems with absorption
Altmann, Eduardo G; Tél, Tamás
2013-01-01
Motivated by applications in optics and acoustics we develop a dynamical-system approach to describe absorption in chaotic systems. We introduce an operator formalism from which we obtain (i) a general formula for the escape rate $\\kappa$ in terms of the natural conditionally-invariant measure of the system; (ii) an increased multifractality when compared to the spectrum of dimensions $D_q$ obtained without taking absorption and return times into account; and (iii) a generalization of the Kantz-Grassberger formula that expresses $D_1$ in terms of $\\kappa$, the positive Lyapunov exponent, the average return time, and a new quantity, the reflection rate. Simulations in the cardioid billiard confirm these results.
Study on Super-Twisting synchronization control of chaotic system based on U model
Directory of Open Access Journals (Sweden)
Jianhua ZHANG
2016-06-01
Full Text Available A U model based Super-Twisting synchronization control method for chaotic systems is proposed. The chaos control of chaotic systems is prescribed, then, based on the current research status of chaotic systems and some useful research results in nonlinear system design, some new methods for chaos control and synchronization are provided, and the controller is designed to achieve the finite time chaos synchronization. The numerical simulations are carried out for Lorenz system and Chen system, and the result proves the effectiveness of the method.
IMPULSIVE CONTROL OF CHAOTIC ATTRACTORS IN NONLINEAR CHAOTIC SYSTEMS
Institute of Scientific and Technical Information of China (English)
马军海; 任彪; 陈予恕
2004-01-01
Based on the study from both domestic and abroad, an impulsive control scheme on chaotic attractors in one kind of chaotic system is presented.By applying impulsive control theory of the universal equation, the asymptotically stable condition of impulsive control on chaotic attractors in such kind of nonlinear chaotic system has been deduced, and with it, the upper bond of the impulse interval for asymptotically stable control was given. Numerical results are presented, which are considered with important reference value for control of chaotic attractors.
Detection of a sudden change of the field time series based on the Lorenz system.
Da, ChaoJiu; Li, Fang; Shen, BingLu; Yan, PengCheng; Song, Jian; Ma, DeShan
2017-01-01
We conducted an exploratory study of the detection of a sudden change of the field time series based on the numerical solution of the Lorenz system. First, the time when the Lorenz path jumped between the regions on the left and right of the equilibrium point of the Lorenz system was quantitatively marked and the sudden change time of the Lorenz system was obtained. Second, the numerical solution of the Lorenz system was regarded as a vector; thus, this solution could be considered as a vector time series. We transformed the vector time series into a time series using the vector inner product, considering the geometric and topological features of the Lorenz system path. Third, the sudden change of the resulting time series was detected using the sliding t-test method. Comparing the test results with the quantitatively marked time indicated that the method could detect every sudden change of the Lorenz path, thus the method is effective. Finally, we used the method to detect the sudden change of the pressure field time series and temperature field time series, and obtained good results for both series, which indicates that the method can apply to high-dimension vector time series. Mathematically, there is no essential difference between the field time series and vector time series; thus, we provide a new method for the detection of the sudden change of the field time series.
Detection of a sudden change of the field time series based on the Lorenz system
Li, Fang; Shen, BingLu; Yan, PengCheng; Song, Jian; Ma, DeShan
2017-01-01
We conducted an exploratory study of the detection of a sudden change of the field time series based on the numerical solution of the Lorenz system. First, the time when the Lorenz path jumped between the regions on the left and right of the equilibrium point of the Lorenz system was quantitatively marked and the sudden change time of the Lorenz system was obtained. Second, the numerical solution of the Lorenz system was regarded as a vector; thus, this solution could be considered as a vector time series. We transformed the vector time series into a time series using the vector inner product, considering the geometric and topological features of the Lorenz system path. Third, the sudden change of the resulting time series was detected using the sliding t-test method. Comparing the test results with the quantitatively marked time indicated that the method could detect every sudden change of the Lorenz path, thus the method is effective. Finally, we used the method to detect the sudden change of the pressure field time series and temperature field time series, and obtained good results for both series, which indicates that the method can apply to high-dimension vector time series. Mathematically, there is no essential difference between the field time series and vector time series; thus, we provide a new method for the detection of the sudden change of the field time series. PMID:28141832
Chaotic systems in optical communications
Siuzdak, J.
2016-09-01
Communications application of chaotic oscillations of lasers with optoelectronic feedback was discussed. The possibility of eavesdropping of the transmission was analyzed. It was proved that if the rogue party precisely knows parameters of the chaotic system it may recreate the entire signals solely by observation of the optical signal power causing security breach.
On closure parameter estimation in chaotic systems
Directory of Open Access Journals (Sweden)
J. Hakkarainen
2012-02-01
Full Text Available Many dynamical models, such as numerical weather prediction and climate models, contain so called closure parameters. These parameters usually appear in physical parameterizations of sub-grid scale processes, and they act as "tuning handles" of the models. Currently, the values of these parameters are specified mostly manually, but the increasing complexity of the models calls for more algorithmic ways to perform the tuning. Traditionally, parameters of dynamical systems are estimated by directly comparing the model simulations to observed data using, for instance, a least squares approach. However, if the models are chaotic, the classical approach can be ineffective, since small errors in the initial conditions can lead to large, unpredictable deviations from the observations. In this paper, we study numerical methods available for estimating closure parameters in chaotic models. We discuss three techniques: off-line likelihood calculations using filtering methods, the state augmentation method, and the approach that utilizes summary statistics from long model simulations. The properties of the methods are studied using a modified version of the Lorenz 95 system, where the effect of fast variables are described using a simple parameterization.
Hybrid internal model control and proportional control of chaotic dynamical systems
Institute of Scientific and Technical Information of China (English)
齐冬莲; 姚良宾
2004-01-01
A new chaos control method is proposed to take advantage of chaos or avoid it. The hybrid Internal Model Control and Proportional Control learning scheme are introduced. In order to gain the desired robust performance and ensure the system's stability, Adaptive Momentum Algorithms are also developed. Through properly designing the neural network plant model and neural network controller, the chaotic dynamical systems are controlled while the parameters of the BP neural network are modified. Taking the Lorenz chaotic system as example, the results show that chaotic dynamical systems can be stabilized at the desired orbits by this control strategy.
Lag synchronization of chaotic systems with time-delayed linear terms via impulsive control
Indian Academy of Sciences (India)
Ranchao Wu; Dongxu Cao
2013-11-01
In this paper, the lag synchronization of chaotic systems with time-delayed linear terms via impulsive control is investigated. Based on the stability theory of impulsive delayed differential equations, some sufficient conditions are obtained guaranteeing the synchronized behaviours between two delayed chaotic systems. Numerical simulations on time-delayed Lorenz and hyperchaotic Chen systems are also carried out to show the effectiveness of the proposed scheme. Note that under the scheme the chaotic system is controlled only at discrete time instants, and so it reduces the control cost in real applications.
Hybrid internal model control and proportional control of chaotic dynamical systems.
Qi, Dong-lian; Yao, Liang-bin
2004-01-01
A new chaos control method is proposed to take advantage of chaos or avoid it. The hybrid Internal Model Control and Proportional Control learning scheme are introduced. In order to gain the desired robust performance and ensure the system's stability, Adaptive Momentum Algorithms are also developed. Through properly designing the neural network plant model and neural network controller, the chaotic dynamical systems are controlled while the parameters of the BP neural network are modified. Taking the Lorenz chaotic system as example, the results show that chaotic dynamical systems can be stabilized at the desired orbits by this control strategy.
Eigenfunctions in chaotic quantum systems
Energy Technology Data Exchange (ETDEWEB)
Baecker, Arnd
2007-07-01
The structure of wavefunctions of quantum systems strongly depends on the underlying classical dynamics. In this text a selection of articles on eigenfunctions in systems with fully chaotic dynamics and systems with a mixed phase space is summarized. Of particular interest are statistical properties like amplitude distribution and spatial autocorrelation function and the implication of eigenfunction structures on transport properties. For systems with a mixed phase space the separation into regular and chaotic states does not always hold away from the semiclassical limit, such that chaotic states may completely penetrate into the region of the regular island. The consequences of this flooding are discussed and universal aspects highlighted. (orig.)
Particle filtering in high-dimensional chaotic systems.
Lingala, Nishanth; Sri Namachchivaya, N; Perkowski, Nicolas; Yeong, Hoong C
2012-12-01
We present an efficient particle filtering algorithm for multiscale systems, which is adapted for simple atmospheric dynamics models that are inherently chaotic. Particle filters represent the posterior conditional distribution of the state variables by a collection of particles, which evolves and adapts recursively as new information becomes available. The difference between the estimated state and the true state of the system constitutes the error in specifying or forecasting the state, which is amplified in chaotic systems that have a number of positive Lyapunov exponents. In this paper, we propose a reduced-order particle filtering algorithm based on the homogenized multiscale filtering framework developed in Imkeller et al. "Dimensional reduction in nonlinear filtering: A homogenization approach," Ann. Appl. Probab. (to be published). In order to adapt the proposed algorithm to chaotic signals, importance sampling and control theoretic methods are employed for the construction of the proposal density for the particle filter. Finally, we apply the general homogenized particle filtering algorithm developed here to the Lorenz'96 [E. N. Lorenz, "Predictability: A problem partly solved," in Predictability of Weather and Climate, ECMWF, 2006 (ECMWF, 2006), pp. 40-58] atmospheric model that mimics mid-latitude atmospheric dynamics with microscopic convective processes.
On the dynamics of a high-order Lorenz-Stenflo system
Rech, Paulo C.
2016-12-01
Results presented in a recent paper in this journal concerning a continuous-time dynamical system, namely that involving high-order Lorenz-Stenflo equations, are extended in this paper. More specifically, the present paper reports on nonlinear dynamics of a six-variable, four-parameter high-order Lorenz-Stenflo system. Six cross-sections of a four-dimensional parameter-space are considered. By using Lyapunov exponents spectra to characterize the dynamical behavior at each point of each of these plots, it is shown that different regions are allowed, from equilibrium point to chaos regions. It is also shown that hyperchaos is not an allowed behavior in a high-order Lorenz-Stenflo system. In addition, new results reported here are compared with those obtained for the original Lorenz-Stenflo system.
Nonlinear control of chaotic systems:A switching manifold approach
Directory of Open Access Journals (Sweden)
Jin-Qing Fang
2000-01-01
Full Text Available In this paper, a switching manifold approach is developed for nonlinear feed-back control of chaotic systems. The design strategy is straightforward, and the nonlinear control law is the simple bang–bang control. Yet, this control method is very effective; for instance, several desired equilibria can be stabilized by using one control law with different initial conditions. Its effectiveness is verified by both theoretical analysis and numerical simulations. The Lorenz system simulation is shown for the purpose of illustration.
Institute of Scientific and Technical Information of China (English)
Liu Ping
2013-01-01
This paper deals with the finite-time stabilization of unified chaotic complex systems with known and unknown parameters.Based on the finite-time stability theory,nonlinear control laws are presented to achieve finite-time chaos control of the determined and uncertain unified chaotic complex systems,respectively.The two controllers are simple,and one of the uncertain unified chaotic complex systems is robust.For the design of a finite-time controller on uncertain unified chaotic complex systems,only some of the unknown parameters need to be bounded.Simulation results for the chaotic complex Lorenz,Lü and Chen systems are presented to validate the design and analysis.
A Novel Concatenated Chaotic Communication System
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
A strategy for a novel concatenated chaotic communication system is presented. The transmitter system comprises chaotic turbo encoder and logistic CSK block in a serially concatenated form. Chaotic turbo code is capable of reducing bit error rate (BER) of the chaotic system in the AWGN channel. Through the chaotic turbo encoder, the coded sequence, which has quasi-chaotic properties, will be transmitted into the logistic CSK block. Having a very sensitive dependence on initial conditions of the map, the logistic CSK block can also be taken as the chaotic authentication method. The receiver, which has logistic demodulation block and chaotic decoder, is a linear asymptotic approximation to the inverse of the transmitter system. A chaotic iterative soft-decision decoding algorithm is also developed based on conventional maximum A posteriori decoding algorithm. At last, a two-step authentication method of this chaotic system is also presented.
A new multi-scroll chaotic system
Institute of Scientific and Technical Information of China (English)
Wang Fa-Qiang; Liu Chong-Xin
2006-01-01
This paper proposes a new simple autonomous chaotic system which can generate multi-scroll chaotic attractors.The characteristic of this new multi-scroll chaotic system is that the 4n + 2m +4-scroll chaotic attractors are generated easily with n and m varying under n ≤ m. Various number of scroll chaotic attractors are illustrated not on ly by computer simulation but also by the realization of an electronic circuit experiment on EWB (Electronics Workbench).
Improving the Complexity of the Lorenz Dynamics
Directory of Open Access Journals (Sweden)
María Pilar Mareca
2017-01-01
Full Text Available A new four-dimensional, hyperchaotic dynamic system, based on Lorenz dynamics, is presented. Besides, the most representative dynamics which may be found in this new system are located in the phase space and are analyzed here. The new system is especially designed to improve the complexity of Lorenz dynamics, which, despite being a paradigm to understand the chaotic dissipative flows, is a very simple example and shows great vulnerability when used in secure communications. Here, we demonstrate the vulnerability of the Lorenz system in a general way. The proposed 4D system increases the complexity of the Lorenz dynamics. The trajectories of the novel system include structures going from chaos to hyperchaos and chaotic-transient solutions. The symmetry and the stability of the proposed system are also studied. First return maps, Poincaré sections, and bifurcation diagrams allow characterizing the global system behavior and locating some coexisting structures. Numerical results about the first return maps, Poincaré cross sections, Lyapunov spectrum, and Kaplan-Yorke dimension demonstrate the complexity of the proposed equations.
Modeling of deterministic chaotic systems
Energy Technology Data Exchange (ETDEWEB)
Lai, Y. [Department of Physics and Astronomy and Department of Mathematics, The University of Kansas, Lawrence, Kansas 66045 (United States); Grebogi, C. [Institute for Plasma Research, University of Maryland, College Park, Maryland 20742 (United States); Grebogi, C.; Kurths, J. [Department of Physics and Astrophysics, Universitaet Potsdam, Postfach 601553, D-14415 Potsdam (Germany)
1999-03-01
The success of deterministic modeling of a physical system relies on whether the solution of the model would approximate the dynamics of the actual system. When the system is chaotic, situations can arise where periodic orbits embedded in the chaotic set have distinct number of unstable directions and, as a consequence, no model of the system produces reasonably long trajectories that are realized by nature. We argue and present physical examples indicating that, in such a case, though the model is deterministic and low dimensional, statistical quantities can still be reliably computed. {copyright} {ital 1999} {ital The American Physical Society}
A chaotic system with only one stable equilibrium
Wang, Xiong; Chen, Guanrong
2012-03-01
If you are given a simple three-dimensional autonomous quadratic system that has only one stable equilibrium, what would you predict its dynamics to be, stable or periodic? Will it be surprising if you are shown that such a system is actually chaotic? Although chaos theory for three-dimensional autonomous systems has been intensively and extensively studied since the time of Lorenz in the 1960s, and the theory has become quite mature today, it seems that no one would anticipate a possibility of finding a three-dimensional autonomous quadratic chaotic system with only one stable equilibrium. The discovery of the new system, to be reported in this Letter, is indeed striking because for a three-dimensional autonomous quadratic system with a single stable node-focus equilibrium, one typically would anticipate non-chaotic and even asymptotically converging behaviors. Although the equilibrium is changed from an unstable saddle-focus to a stable node-focus, therefore the familiar Ši'lnikov homoclinic criterion is not applicable, it is demonstrated to be chaotic in the sense of having a positive largest Lyapunov exponent, a fractional dimension, a continuous broad frequency spectrum, and a period-doubling route to chaos.
Chaotic Synchronzation System and Electrocardiogram
Institute of Scientific and Technical Information of China (English)
LiuqingPei; XinlaiDai; 等
1997-01-01
A mathematical model of chaotic synchronization of the heart-blood flow coupling dynamics is propsed,which is based on a seven dimension nonlinear dynamical system constructed by three subsystems of the sinoatrial node natural pacemaker,the cardiac relaxation oscillator and the dynamics of blood-fluid in heart chambers.The existence and robustness of the self-chaotic synchronization of the system are demonstrated by both methods of theoretical analysis and numerical simulation.The spectrum of Lyapunov exponent,the Lyapunov dimension and the Kolmogorov entropy are estimated when the system was undergoing the state of self-chaotic synchronization evolution.The time waveform of the dynamical variable,which represents the membrane potential of the cardiac integrative cell,shows a shape which is similar to that of the normal electrocardiogram(ECG) of humans,thus implying that the model possesses physiological significance functionally.
Adaptive control and synchronization of an uncertain new hyperchaotic Lorenz system
Institute of Scientific and Technical Information of China (English)
Cai Guo-Liang; Zheng Song; Tian Li-Xin
2008-01-01
This paper is involved with the adaptive control and synchronization problems for an uncertain new hyperchaotic Lorenz system. Based on the Lyapunov stability theory, the adaptive control law is derived such that the trajectory of hyperchaotic Lorenz system with unknown parameters can be globally stabilized to an unstable equilibrium point of the uncontrolled system. Furthermore, an adaptive control approach is presented to the synchronizations between two identical hyperchaotic systems, particularly between two different uncertain hyperchaotic systems. Numerical simulations show the effectiveness of the presented method.
Institute of Scientific and Technical Information of China (English)
WANG Qi
2006-01-01
In this paper, a bidirectional partial generalized (lag, complete, and anticipated) synchronization of a class of continuous-time systems is defined. Then based on the active control idea, a new systematic and concrete scheme is developed to achieve bidirectional partial generalized (lag, complete, and anticipated) synchronization between two chaotic systems or between chaotic and hyperchaotic systems. With the help of symbolic-numerical computation,we choose the modified Chua system, Lorenz system, and the hyperchaotic Tamasevicius-Namajunas-Cenys system to illustrate the proposed scheme. Numerical simulations are used to verify the effectiveness of the proposed scheme. It is interesting that partialchaos synchronization not only can take place between two chaotic systems, but also can take place between chaotic and hyperchaotic systems. The proposed scheme can also be extended to research bidirectional partial generalized (lag, complete, and anticipated) synchronization between other dynamical systems.
Dynamical properties and complexity in fractional-order diffusionless Lorenz system
He, Shaobo; Sun, Kehui; Banerjee, Santo
2016-08-01
In this paper, dynamics and complexity of the fractional-order diffusionless Lorenz system which is solved by the developed discrete Adomian decomposition method are investigated numerically. Dynamical properties of the fractional-order diffusionless Lorenz system with the control parameter and derivative order varying is analyzed by using bifurcation diagrams, and period-doubling route to chaos in different cases is observed. The complexity of the system is investigated by means of Lyapunov characteristic exponents, multi-scale spectral entropy algorithm and multiscale Renyi permutation entropy algorithm. It can be observed that the three methods illustrate consistent results and the system has rich complex dynamics. Interestingly, complexity decreases with the increase of derivative order. It shows that the fractional-order diffusionless Lorenz system is a good model for real applications such as information encryption and secure communication.
Chaotic Dynamics in Hybrid Systems
P.J. Collins (Pieter)
2008-01-01
htmlabstractIn this paper we give an overview of some aspects of chaotic dynamics in hybrid systems, which comprise different types of behaviour. Hybrid systems may exhibit discontinuous dependence on initial conditions leading to new dynamical phenomena. We indicate how methods from topological
Chaotic dynamics in hybrid systems
P.J. Collins (Pieter)
2008-01-01
htmlabstractIn this paper we give an overview of some aspects of chaotic dynamics in hybrid systems, which comprise different types of behaviour. Hybrid systems may exhibit discontinuous dependence on initial conditions leading to new dynamical phenomena. We indicate how methods from topological
Lectures on chaotic dynamical systems
Afraimovich, Valentin
2002-01-01
This book is devoted to chaotic nonlinear dynamics. It presents a consistent, up-to-date introduction to the field of strange attractors, hyperbolic repellers, and nonlocal bifurcations. The authors keep the highest possible level of "physical" intuition while staying mathematically rigorous. In addition, they explain a variety of important nonstandard algorithms and problems involving the computation of chaotic dynamics. The book will help readers who are not familiar with nonlinear dynamics to understand and appreciate sophisticated modern dynamical systems and chaos. Intended for courses in either mathematics, physics, or engineering, prerequisites are calculus, differential equations, and functional analysis.
Chaotic and Arnold stripes in weakly chaotic Hamiltonian systems.
Custódio, M S; Manchein, C; Beims, M W
2012-06-01
The dynamics in weakly chaotic Hamiltonian systems strongly depends on initial conditions (ICs) and little can be affirmed about generic behaviors. Using two distinct Hamiltonian systems, namely one particle in an open rectangular billiard and four particles globally coupled on a discrete lattice, we show that in these models, the transition from integrable motion to weak chaos emerges via chaotic stripes as the nonlinear parameter is increased. The stripes represent intervals of initial conditions which generate chaotic trajectories and increase with the nonlinear parameter of the system. In the billiard case, the initial conditions are the injection angles. For higher-dimensional systems and small nonlinearities, the chaotic stripes are the initial condition inside which Arnold diffusion occurs.
Communication Scheme via Cascade Chaotic Systems
Institute of Scientific and Technical Information of China (English)
HUA Chang-Chun; GUAN Xin-Ping
2004-01-01
@@ A new chaotic communication scheme is constructed. Different from the existing literature, cascade chaotic systems are employed. Two cascade modes are considered. First, we investigate the input to state cascade mode;cascade systems between different kinds of chaotic systems are considered. Then the parameter cascade case of chaotic system is studied. Under the different cases, the corresponding receivers are designed, which can succeed in recovering the former emitted signal. Simulations are performed to verify the validity of the proposed main results.
Generalized reduced-order synchronization of chaotic system based on fast slide mode
Institute of Scientific and Technical Information of China (English)
Gao Tie-Gang; Chen Zeng-Qiang; Yuan Zhu-Zhi
2005-01-01
A new kind of generalized reduced-order synchronization of different chaotic systems is proposed in this paper.It is shown that dynamical evolution of third-order oscillator can be synchronized with the canonical projection of a fourth-order chaotic system generated through nonsingular states transformation from a cell neural net chaotic system.In this sense, it is said that generalized synchronization is achieved in reduced-order. The synchronization discussed here expands the scope of reduced-order synchronization studied in relevant literatures. In this way, we can achieve generalized reduced-order synchronization between many famous chaotic systems such as the second-order D(u)ffing system and the third-order Lorenz system by designing a fast slide mode controller. Simulation results are provided to verify the operation of the designed synchronization.
Observers for a Class of Chaotic Systems
Institute of Scientific and Technical Information of China (English)
ZHOU Ping
2006-01-01
The design of observers for a class of practical physical chaotic systems is discussed.By using only one state variable and its time derivatives,a control law is constructed to achieve the synchronization between the investigated chaotic systems and their observers,and the results are proved theoretically.Several observers of chaotic systems are designed by using this method.
Parameter estimation for chaotic systems based on improved boundary chicken swarm optimization
Chen, Shaolong; Yan, Renhuan
2016-10-01
Estimating unknown parameters for chaotic system is a key problem in the field of chaos control and synchronization. Through constructing an appropriate fitness function, parameter estimation of chaotic system could be converted to a multidimensional parameter optimization problem. In this paper, a new method base on improved boundary chicken swarm optimization (IBCSO) algorithm is proposed for solving the problem of parameter estimation in chaotic system. However, to the best of our knowledge, there is no published research work on chicken swarm optimization for parameters estimation of chaotic system. Computer simulation based on Lorenz system and comparisons with chicken swarm optimization, particle swarm optimization, and genetic algorithm shows the effectiveness and feasibility of the proposed method.
Estimating the globally attractive set and positively invariant set of a unified chaotic system
Institute of Scientific and Technical Information of China (English)
SHU Yong-lu; ZHANG Yong-hao
2008-01-01
By constructing two suitable generalized Lyapunov functions, we derived a generalized ellipsoidal estimate of the globally attractive set and positively invariant set of the unified chaotic system with the parameters a=1/29 and 1/29Lorenz system and a unified chaotic system, Journal of Mathematical Analysis and Applications, 2006, 323(2): 844-853]. The theoretical results obtained in this paper will find wide application in chaos control and synchronization.
Advances and applications in chaotic systems
Volos, Christos
2016-01-01
This book reports on the latest advances and applications of chaotic systems. It consists of 25 contributed chapters by experts who are specialized in the various topics addressed in this book. The chapters cover a broad range of topics of chaotic systems such as chaos, hyperchaos, jerk systems, hyperjerk systems, conservative and dissipative systems, circulant chaotic systems, multi-scroll chaotic systems, finance chaotic system, highly chaotic systems, chaos control, chaos synchronization, circuit realization and applications of chaos theory in secure communications, mobile robot, memristors, cellular neural networks, etc. Special importance was given to chapters offering practical solutions, modeling and novel control methods for the recent research problems in chaos theory. This book will serve as a reference book for graduate students and researchers with a basic knowledge of chaos theory and control systems. The resulting design procedures on the chaotic systems are emphasized using MATLAB software.
Synchronization of indirectly coupled Lorenz oscillators: An experimental study
Indian Academy of Sciences (India)
Amit Sharma; Manish Dev Shrimali
2011-11-01
The dynamics of indirectly coupled Lorenz circuits is investigated experimentally. The in-phase and anti-phase synchronization of indirectly coupled chaotic oscillators reported in Phys. Rev. E 81, 046216 (2010) is veriﬁed by physical experiments with electronic circuits. Two chaotic systems coupled through a common dynamic environment shows the verity of synchronization behaviours such as anti-phase synchronization, in-phase synchronization, identical synchronization, anti-synchronization, etc.
Perfect synchronization of chaotic systems: a controllability perspective
Institute of Scientific and Technical Information of China (English)
Sun Ming-Xuan; He Xiong-Xiong; Yu Li
2006-01-01
This paper presents a synchronization method, motivated from the constructive controllability analysis, for two identical chaotic systems. This technique is applied to achieve perfect synchronization for Lorenz systems and coupled dynamo systems. It turns out that states of the drive system and the response system are synchronized within finite time, and the reaching time is independent of initial conditions, which can be specified in advance. In addition to the simultaneous synchronization, the response system is synchronized un-simultaneously to the drive system with different reaching time for each state. The performance of the resulting system is analytically quantified in the face of initial condition error, and with numerical experiments the proposed method is demonstrated to perform well.
A chaotic system with only one stable equilibrium
Xiong, Wang
2011-01-01
If you are given a simple three-dimensional autonomous quadratic system that has only one stable equilibrium, what would you predict its dynamics to be, stable or periodic? Will it be surprising if you are shown that such a system is actually chaotic? Although chaos theory for three-dimensional autonomous systems has been intensively and extensively studied since the time of Lorenz in the 1960s, and the theory has become quite mature today, it seems that no one would anticipate a possibility of finding a three-dimensional autonomous quadratic chaotic system with only one stable equilibrium. The discovery of the new system, to be reported in this Letter, is indeed striking because for a three-dimensional autonomous quadratic system with a single stable node-focus equilibrium, one typically would anticipate non-chaotic and even asymptotically converging behaviors. Although the new system is not of saddle-focus type, therefore the familiar \\v{S}i'lnikov homoclinic criterion is not applicable, it is demonstrated ...
Institute of Scientific and Technical Information of China (English)
Wang Dong-Feng; Zhang Jin-Ying; Wang Xiao-Yan
2013-01-01
This paper provides a novel method to synchronize uncertain fractional-order chaotic systems with external disturbance via fractional terminal sliding mode control.Based on Lyapunov stability theory,a new fractional-order switching manifold is proposed,and in order to ensure the occurrence of sliding motion in finite time,a corresponding sliding mode control law is designed.The proposed control scheme is applied to synchronize the fractional-order Lorenz chaotic system and fractional-order Chen chaotic system with uncertainty and external disturbance parameters.The simulation results show the applicability and efficiency of the proposed scheme.
Impulsive Synchronization of Discrete Chaotic Systems
Institute of Scientific and Technical Information of China (English)
郑永爱; 年漪蓓; 刘曾荣
2003-01-01
Impulsive synchronization of two chaotic maps is reformulated as impulsive control of the synchronization error system. We then present a theorem on the asymptotic synchronization of two chaotic maps by using synchronization impulses with varying impulsive intervals. As an example and application of the theorem, we derives some sufficient conditions for the synchronization of two chaotic Lozi maps via impulsive control. The effectiveness of this approach has been demonstrated with chaotic Lozi map.
Institute of Scientific and Technical Information of China (English)
Si Gang-Quan; Sun Zhi-Yong; Zhang Yan-Bin
2011-01-01
This paper investigates the synchronization between integer-order and fractional-order chaotic systems.By introducing fractional-order operators into the controllers,the addressed problem is transformed into a synchronization one among integer-order systems.A novel general method is presented in the paper with rigorous proof.Based on this method,effective controllers are designed for the synchronization between Lorenz systems with an integer order and a fractional order,and for the synchronization between an integer-order Chen system and a fractional-order Liu system.Numerical results,which agree well with the theoretical analyses,are also given to show the effectiveness of this method.
A new reduced-order observer design for the synchronization of Lorenz systems
Energy Technology Data Exchange (ETDEWEB)
Martinez-Guerra, R. [Departamento de Control Automatico, CINVESTAV-IPN, AP 14-740, CP 07360, Mexico, DF (Mexico)] e-mail: rguerra@ctrl.cinvestav.mx; Cruz-Victoria, J.C. [Departamento de Control Automatico, CINVESTAV-IPN, AP 14-740, CP 07360, Mexico, DF (Mexico); Gonzalez-Galan, R. [Departamento de Control Automatico, CINVESTAV-IPN, AP 14-740, CP 07360, Mexico, DF (Mexico); Aguilar-Lopez, R. [Departamento de Energia, UAM-Azcapotzalco, 02200 (Mexico)
2006-04-01
In this paper we tackle the synchronization of Lorenz system problem using a new proportional reduced-order observer design in the algebraic and differential setting. We prove the asymptotic stability of the resulting error system and by means of algebraic manipulations we obtain the estimates of the current states (master system), the construction of a proportional reduced-order observer is the main ingredient in our approach. Finally, we present a simulation to illustrate the effectiveness of the suggested approach.
Output Regulation of the Arneodo Chaotic System
Sundarapandian Vaidyanathan
2010-01-01
This paper solves the problem of regulating the output of the Arneodo chaotic system (1981), which is one of the paradigms of chaotic dynamical systems. Explicitly, using the state feedback control laws, the output of the Arneodo chaotic system is regulated so as to track constant reference signals as well as to track periodic reference signals. The control laws are derived using the regulator equations of Byrnes and Isidori (1990), which provide the solution of the output regulation problem ...
Robust synchronization of chaotic systems via feedback
Energy Technology Data Exchange (ETDEWEB)
Femat, Ricardo [IPICYT, San Luis Potosi (Mexico). Dept. de Matematicas Aplicadas; Solis-Perales, Gualberto [Universidad de Guadalajara, Centro Univ. de Ciencias Exactas e Ingenierias (Mexico). Div. de Electronica y Computacion
2008-07-01
This volume includes the results derived during last ten years about both suppression and synchronization of chaotic -continuous time- systems. Along this time, the concept was to study how the intrinsic properties of dynamical systems can be exploited to suppress and to synchronize the chaotic behaviour and what synchronization phenomena can be found under feedback interconnection. A compilation of these findings is described in this book. This book shows a perspective on synchronization of chaotic systems. (orig.)
Identification of fractional chaotic system parameters
Energy Technology Data Exchange (ETDEWEB)
Al-Assaf, Yousef E-mail: yassaf@aus.ac.ae; El-Khazali, Reyad E-mail: khazali@ece.ac.ae; Ahmad, Wajdi E-mail: wajdi@sharjah.ac.ae
2004-11-01
In this work, a technique is introduced for parameter identification of fractional order chaotic systems. Features are extracted, from chaotic system outputs obtained for different system parameters, using discrete Fourier transform (DFT), power spectral density (PSD), and wavelets transform (WT). Artificial neural networks (ANN) are then trained on these features to predict the fractional chaotic system parameters. A fractional chaotic oscillator model is used through this work to demonstrate the developed technique. Numerical results show that recurrent Jordan-Elman neural networks with features obtained by the PSD estimate via Welch functions give adequate identification accuracy compared to other techniques.
Chaotic evolution of the solar system
Sussman, Gerald J.; Wisdom, Jack
1992-01-01
The evolution of the entire planetary system has been numerically integrated for a time span of nearly 100 million years. This calculation confirms that the evolution of the solar system as a whole is chaotic, with a time scale of exponential divergence of about 4 million years. Additional numerical experiments indicate that the Jovian planet subsystem is chaotic, although some small variations in the model can yield quasi-periodic motion. The motion of Pluto is independently and robustly chaotic.
The transient ladder synchronization of chaotic systems
Energy Technology Data Exchange (ETDEWEB)
Chen, H.-K. [Department of Industrial Engineering and Management, Hsiuping Institute of Technology, No. 11, Gungye Rd., Dali City, Taichung, Taiwan (China)]. E-mail: kanechen@giga.net.tw; Sheu, L.-J. [Department of Mechanical Engineering, Chung Hua University, Hsinchu, Taiwan (China)]. E-mail: ljsheu@chu.edu.tw
2006-07-03
A new type for chaotically synchronizing systems, transient ladder chaos synchronization, is proposed in this Letter. For some physical systems, chaotic synchronization is possible in only some of the variables. It is shown that, for the non-synchronizing variable, synchronization up to a constant difference for t{sub 1}=
Sensitivity analysis on chaotic dynamical system by Non-Intrusive Least Square Shadowing (NILSS)
Ni, Angxiu
2016-01-01
This paper develops the tangent Non-Intrusive Least Square Shadowing (NILSS) method, which computes sensitivity for chaotic dynamical systems. In NILSS, a tangent solution is represented as a linear combination of a inhomogeneous tangent solution and some homogeneous tangent solutions. Then we solve a least square problem under this new representation. As a result, this new variant is easier to implement with existing solvers. For chaotic systems with large degrees of freedom but low dimensional attractors, NILSS has low computation cost. NILSS is applied to two chaotic PDE systems: the Lorenz 63 system, and a CFD simulation of a backward-facing step. The results show that NILSS computes the correct derivative with a lower cost than the conventional Least Square Shadowing method and the conventional finite difference method.
Institute of Scientific and Technical Information of China (English)
ZHOU Qian; CHEN Zeng-Qiang; YUAN Zhu-Zhi
2008-01-01
Blowout bifurcation in nonlinear systems occurs when a chaotic attractor lying in some symmetric subspace becomes transversely unstable. A class of five-dimensional continuous autonomous systems is considered, in which a two-dimensional subsystem is driven by a family of generalized Lorenz systems. The systems have some common dynamical characters. As the coupling parameter changes, blowout bifurcations occur in these systems and brings on change of the systems' dynamics. After the bifurcation the phenomenon of on-ff intermittency appears. It is observed that the systems undergo a symmetric hyperchaos-chaos hyperchaos transition via or after blowout bifurcations. An example of the systems is given, in which the drive system is the Chen system. We investigate the dynamical behaviour before and after the blowout bifurcation in the systems and make an analysis of the transition process. It is shown that in such coupled chaotic continuous systems, blowout bifurcation leads to a transition from chaos to hyperchaos for the whole systems, which provides a route to hyperchaos.
Directory of Open Access Journals (Sweden)
Jun Wang
2016-01-01
Full Text Available This paper proposes an improved cuckoo search (ICS algorithm to establish the parameters of chaotic systems. In order to improve the optimization capability of the basic cuckoo search (CS algorithm, the orthogonal design and simulated annealing operation are incorporated in the CS algorithm to enhance the exploitation search ability. Then the proposed algorithm is used to establish parameters of the Lorenz chaotic system and Chen chaotic system under the noiseless and noise condition, respectively. The numerical results demonstrate that the algorithm can estimate parameters with high accuracy and reliability. Finally, the results are compared with the CS algorithm, genetic algorithm, and particle swarm optimization algorithm, and the compared results demonstrate the method is energy-efficient and superior.
Wang, Jun; Zhou, Bihua; Zhou, Shudao
2016-01-01
This paper proposes an improved cuckoo search (ICS) algorithm to establish the parameters of chaotic systems. In order to improve the optimization capability of the basic cuckoo search (CS) algorithm, the orthogonal design and simulated annealing operation are incorporated in the CS algorithm to enhance the exploitation search ability. Then the proposed algorithm is used to establish parameters of the Lorenz chaotic system and Chen chaotic system under the noiseless and noise condition, respectively. The numerical results demonstrate that the algorithm can estimate parameters with high accuracy and reliability. Finally, the results are compared with the CS algorithm, genetic algorithm, and particle swarm optimization algorithm, and the compared results demonstrate the method is energy-efficient and superior.
Chaotic control and synchronization for system identification.
Carroll, T L
2004-04-01
Research into applications of synchronized chaotic systems assumes that it will be necessary to build many different drive-response pairs, but little is known in general about designing higher dimensional chaotic flows. In this paper, I do not add any design techniques, but I show that it is possible to create multiple drive-response pairs from one chaotic system by applying chaos control techniques to the drive and response systems. If one can design one chaotic system with the desired properties, then many drive-response pairs can be built from this system, so that it is not necessary to solve the design problem more than once. I show both numerical simulations and experimental work with chaotic circuits. I also test the response systems for ability to overcome noise or other interference.
Identification of Chaotic Systems with Application to Chaotic Communication
Institute of Scientific and Technical Information of China (English)
FENG Jiu-Chao; QIU Yu-Hui
2004-01-01
@@ We propose and develop a novel method to identify a chaotic system with time-varying bifurcation parameters via an observation signal which has been contaminated by additive white Gaussian noise. This method is based on an adaptive algorithm, which takes advantage of the good approximation capability of the radial basis function neural network and the ability of the extended Kalman filter for tracking a time-varying dynamical system. It is demonstrated that, provided the bifurcation parameter varies slowly in a time window, a chaotic dynamical system can be tracked and identified continuously, and the time-varying bifurcation parameter can also be retrieved in a sub-window of time via a simple least-square-fit method.
Energy Technology Data Exchange (ETDEWEB)
Marino, Ines P. [Nonlinear Dynamics and Chaos Group, Departamento de Matematicas y Fisica Aplicadas y Ciencias de la Naturaleza, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid (Spain)]. E-mail: ines.perez@urjc.es; Miguez, Joaquin [Departamento de Teoria de la Senal y Comunicaciones, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganes, Madrid (Spain)]. E-mail: jmiguez@ieee.org
2006-03-06
We address the problem of estimating the unknown parameters of a primary chaotic system that produces an observed time series. These observations are used to drive a secondary system in a way that ensures synchronization when the two systems have identical parameters. We propose a new method to adaptively adjust the parameters in the secondary system until synchronization is achieved. It is based on the gradient-descent optimization of a suitably defined cost function and can be systematically applied to arbitrary systems. We illustrate its application by estimating the complete parameter vector of a Lorenz system.
Synchronization Techniques for Chaotic Communication Systems
Jovic, Branislav
2011-01-01
Since the early 1990s, when synchronization of chaotic communication systems became a popular research subject, a vast number of scientific papers have been published. However, most of today's books on chaotic communication systems deal exclusively with the systems where perfect synchronization is assumed, an assumption which separates theoretical from practical, real world, systems. This book is the first of its kind dealing exclusively with the synchronization techniques for chaotic communication systems. It describes a number of novel robust synchronization techniques, which there is a lack
Output Regulation of the Arneodo Chaotic System
Directory of Open Access Journals (Sweden)
Sundarapandian Vaidyanathan
2010-08-01
Full Text Available This paper solves the problem of regulating the output of the Arneodo chaotic system (1981, which is one of the paradigms of chaotic dynamical systems. Explicitly, using the state feedback control laws, the output of the Arneodo chaotic system is regulated so as to track constant reference signals as well as to track periodic reference signals. The control laws are derived using the regulator equations of Byrnes and Isidori (1990, which provide the solution of the output regulation problem for nonlinear control systems involving neutrally stable exosystem dynamics. Numerical results are shown to verify the results.
Generation and Nonlinear Dynamical Analyses of Fractional-Order Memristor-Based Lorenz Systems
Directory of Open Access Journals (Sweden)
Huiling Xi
2014-11-01
Full Text Available In this paper, four fractional-order memristor-based Lorenz systems with the flux-controlled memristor characterized by a monotone-increasing piecewise linear function, a quadratic nonlinearity, a smooth continuous cubic nonlinearity and a quartic nonlinearity are presented, respectively. The nonlinear dynamics are analyzed by using numerical simulation methods, including phase portraits, bifurcation diagrams, the largest Lyapunov exponent and power spectrum diagrams. Some interesting phenomena, such as inverse period-doubling bifurcation and intermittent chaos, are found to exist in the proposed systems.
Chaos Control and Synchronization in Fractional-Order Lorenz-Like System
Directory of Open Access Journals (Sweden)
Sachin Bhalekar
2012-01-01
Full Text Available The present paper deals with fractional-order version of a dynamical system introduced by Chongxin et al. (2006. The chaotic behavior of the system is studied using analytic and numerical methods. The minimum effective dimension is identified for chaos to exist. The chaos in the proposed system is controlled using simple linear feedback controller. We design a controller to place the eigenvalues of the system Jacobian in a stable region. The effectiveness of the controller in eliminating the chaotic behavior from the state trajectories is also demonstrated using numerical simulations. Furthermore, we synchronize the system using nonlinear feedback.
Adaptive synchronization of T-S fuzzy chaotic systems with unknown parameters
Energy Technology Data Exchange (ETDEWEB)
Kim, Jae-Hun [Department of Electrical and Electronic Engineering, Yonsei University, 134 Shinchon-dong, Sudaemoon-gu, Seoul 120-749 (Korea, Republic of)]. E-mail: jhkim@yeics.yonsei.ac.kr; Park, Chang-Woo [Precision Machinery Research Center, Korea Electronics Technology Institute, 203-103 B/D 192, Yakdae-dong, Wonmi-gu, Puchon-si, Kyunggi-do 420-140 (Korea, Republic of); Kim, Euntai [Department of Electrical and Electronic Engineering, Yonsei University, 134 Shinchon-dong, Sudaemoon-gu, Seoul 120-749 (Korea, Republic of); Park, Mignon [Department of Electrical and Electronic Engineering, Yonsei University, 134 Shinchon-dong, Sudaemoon-gu, Seoul 120-749 (Korea, Republic of)
2005-06-01
This paper presents a fuzzy model-based adaptive approach for synchronization of chaotic systems which consist of the drive and response systems. Takagi-Sugeno (T-S) fuzzy model is employed to represent the chaotic drive and response systems. Since the parameters of the drive system are assumed unknown, we design the response system that estimates the parameters of the drive system by adaptive strategy. The adaptive law is derived to estimate the unknown parameters and its stability is guaranteed by Lyapunov stability theory. In addition, the controller in the response system contains two parts: one part that can stabilize the synchronization error dynamics and the other part that estimates the unknown parameters. Numerical examples, including Duffing oscillator and Lorenz attractor, are given to demonstrate the validity of the proposed adaptive synchronization approach.
Multivariate permutation entropy and its application for complexity analysis of chaotic systems
He, Shaobo; Sun, Kehui; Wang, Huihai
2016-11-01
To measure the complexity of multivariate systems, the multivariate permutation entropy (MvPE) algorithm is proposed. It is employed to measure complexity of multivariate system in the phase space. As an application, MvPE is applied to analyze the complexity of chaotic systems, including hyperchaotic Hénon map, fractional-order simplified Lorenz system and financial chaotic system. Results show that MvPE algorithm is effective for analyzing the complexity of the multivariate systems. It also shows that fractional-order system does not become more complex with derivative order varying. Compared with PE, MvPE has better robustness for noise and sampling interval, and the results are not affected by different normalization methods.
Chaotic dynamics of controlled electric power systems
Kozlov, V. N.; Trosko, I. U.
2016-12-01
The conditions for appearance of chaotic dynamics of electromagnetic and electromechanical processes in energy systems described by the Park-Gorev bilinear differential equations with account for lags of coordinates and restrictions on control have been formulated. On the basis of classical equations, the parameters of synchronous generators and power lines, at which the chaotic dynamics of energy systems appears, have been found. The qualitative and quantitative characteristics of chaotic processes in energy associations of two types, based on the Hopf theorem, and methods of nonstationary linearization and decompositions are given. The properties of spectral characteristics of chaotic processes have been investigated, and the qualitative similarity of bilinear equations of power systems and Lorentz equations have been found. These results can be used for modernization of the systems of control of energy objects. The qualitative and quantitative characteristics for power energy systems as objects of control and for some laws of control with the feedback have been established.
Fast, parallel and secure cryptography algorithm using Lorenz's attractor
Marco, Anderson Gonçalves; Bruno, Odemir Martinez; 10.1142/S0129183110015166
2012-01-01
A novel cryptography method based on the Lorenz's attractor chaotic system is presented. The proposed algorithm is secure and fast, making it practical for general use. We introduce the chaotic operation mode, which provides an interaction among the password, message and a chaotic system. It ensures that the algorithm yields a secure codification, even if the nature of the chaotic system is known. The algorithm has been implemented in two versions: one sequential and slow and the other, parallel and fast. Our algorithm assures the integrity of the ciphertext (we know if it has been altered, which is not assured by traditional algorithms) and consequently its authenticity. Numerical experiments are presented, discussed and show the behavior of the method in terms of security and performance. The fast version of the algorithm has a performance comparable to AES, a popular cryptography program used commercially nowadays, but it is more secure, which makes it immediately suitable for general purpose cryptography ...
Li, Xiang-Tao; Yin, Ming-Hao
2012-05-01
We study the parameter estimation of a nonlinear chaotic system, which can be essentially formulated as a multidimensional optimization problem. In this paper, an orthogonal learning cuckoo search algorithm is used to estimate the parameters of chaotic systems. This algorithm can combine the stochastic exploration of the cuckoo search and the exploitation capability of the orthogonal learning strategy. Experiments are conducted on the Lorenz system and the Chen system. The proposed algorithm is used to estimate the parameters for these two systems. Simulation results and comparisons demonstrate that the proposed algorithm is better or at least comparable to the particle swarm optimization and the genetic algorithm when considering the quality of the solutions obtained.
Energy Technology Data Exchange (ETDEWEB)
Zheng Yongai, E-mail: zhengyongai@163.co [Department of Computer, Yangzhou University, Yangzhou, 225009 (China); Nian Yibei [School of Energy and Power Engineering, Yangzhou University, Yangzhou, 225009 (China); Wang Dejin [Department of Computer, Yangzhou University, Yangzhou, 225009 (China)
2010-12-01
In this Letter, a kind of novel model, called the generalized Takagi-Sugeno (T-S) fuzzy model, is first developed by extending the conventional T-S fuzzy model. Then, a simple but efficient method to control fractional order chaotic systems is proposed using the generalized T-S fuzzy model and adaptive adjustment mechanism (AAM). Sufficient conditions are derived to guarantee chaos control from the stability criterion of linear fractional order systems. The proposed approach offers a systematic design procedure for stabilizing a large class of fractional order chaotic systems from the literature about chaos research. The effectiveness of the approach is tested on fractional order Roessler system and fractional order Lorenz system.
SPECIAL DYNAMIC BEHAVIORS OF A TEMPORAL CHAOTIC SYSTEM
Institute of Scientific and Technical Information of China (English)
Mingxuan Zhang; Jinjiang Yu; Wangqiang Han
2008-01-01
When dynamic behaviors of temporal chaotic system are analyzed,we find that a temporal chaotic system has not only genetic dynamic behaviors of chaotic reflection,but also has phenomena influencing two chaotic attractors by original values.Along with the system parameters changing to certain value,the system will appear a break in chaotic region,and jump to another orbit of attractors.When it is opposite that the system parameters change direction,the temporal chaotic system appears complicated chaotic behaviors.
Image encryption using high-dimension chaotic system
Institute of Scientific and Technical Information of China (English)
Sun Fu-Yan; Liu Shu-Tang; Lü Zong-Wang
2007-01-01
In recent years, the chaos based cryptographic algorithms have suggested some new and efficient ways to develop secure image encryption techniques. This paper proposes a new approach for image encryption based on a highdimensional chaotic map. The new scheme employs the Cat map to shuffle the positions, then to confuse the relationship between the cipher-image and the plain-image using the high-dimensional Lorenz chaotic map preprocessed. The results of experimental, statistical analysis and key space analysis show that the proposed image encryption scheme provides an efficient and secure way for real-time image encryption and transmission.
Adaptive tracking control of chaotic systems
Institute of Scientific and Technical Information of China (English)
卢钊; 卢和
2004-01-01
It is important to develop control techniques able to control not only known chaos but also chaotic systems with unknown parameters. This paper proposes a novel adaptive tracking control approach for identifying the unknown parameters and controlling the chaos, which is not closely related to the particular chaotic system to be controlled. The global uniform boundedness of estimated parameters and the asymptotical stability of the tracking errors are proved by Lyapunov stability theory and LaSalle-Yoshizawa theorem. The suggested method enables stabilization of chaotic motion to a steady state ad well as tracking of any desired trajectory to be achieved in a systematic way. Computer simulation on a complex chaotic system illustrtes the effectiveness of the proposed control method.
Synchronization of chaotic systems with parameter driven by a chaotic signal
Energy Technology Data Exchange (ETDEWEB)
Li Guohui [Department of Communication Engineering, Shanghai University, Yanchang Road 149, Shanghai 200072 (China)] e-mail: ghlee@shl63.net
2005-12-01
Chaos control with driving parameter scheme in uncoupled identical chaotic oscillators is presented. By driving the parameter of chaotic systems using external chaotic signal, synchronization and anti-synchronization can be implemented. Numerical simulations show that either synchronization or anti-synchronization can appear depending significantly on initial condition and on driving strength. The proposed method is particularly suited for a variety of chaotic systems, which cannot couple with each other in engineering.
Theoretical Basis and Application of an Analogue-Dynamical Model in the Lorenz System
Institute of Scientific and Technical Information of China (English)
REN Hongli; CHOU Jifan; HUANG Jianping; ZHANG Peiqun
2009-01-01
The theoretical basis and application of an analogue-dynamical model (ADM) in the Lorenz system is studied. The ADM can effectively combine statistical and dynamical methods in which the small disturbance of the current initial value superimposed on the historical analogue reference state can be regarded as a prediction objective. Primary analyses show that under the condition of appending disturbances in model parameters, the model errors of ADM are much smaller than those of the pure dynamical model (PDM).The characteristics of predictability on the ADM in the Lorenz system are analyzed in phase space by conducting case studies and global experiments. The results show that the ADM can quite effectively reduce prediction errors and prolong the valid time of the prediction in most situations in contrast to the PDM, but when model errors are considerably small, the latter will be superior to the former. To overcome such a problem, the multi-reference-state updating can be applied to introduce the information of multi-analogue and update analogue and can exhibit exciting performance in the ADM.
Fuzzy modeling and synchronization of hyper chaotic systems
Energy Technology Data Exchange (ETDEWEB)
Zhang Hongbin [Center for Nonlinear and Complex Systems, School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 610054 (China)] e-mail: zhanghb@uestc.edu.cn; Liao Xiaofeng [Center for Nonlinear and Complex Systems, School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 610054 (China); Institute of Computer Science, Chongqing University, Chongqing 400044 (China); Yu Juebang [Center for Nonlinear and Complex Systems, School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 610054 (China)
2005-11-01
This paper presents fuzzy model-based designs for synchronization of hyper chaotic systems. The T-S fuzzy models for hyper chaotic systems are exactly derived. Based on the T-S fuzzy hyper chaotic models, the fuzzy controllers for hyper chaotic synchronization are designed via the exact linearization techniques. Numerical examples are given to demonstrate the effectiveness of the proposed method.
CHAOTIC CONTROL OF NONLINEAR SYSTEMS BASED ON IMPROVED CORRELATIVITY
Institute of Scientific and Technical Information of China (English)
Zhou Xiaoan; Zhang Jihong
2003-01-01
Chaotic sequences are basically ergodic random sequences. By improving correlativity of a chaotic signal, the chaotic dynamic system can be controlled to converge to its equilibrium point and, more significantly, to its multi-periodic orbits. Mathematical theory analysis is carried out and some computer simulation results are provided to support such controllability of the chaotic Henon system and the discrete coupled map lattice.
Multiple channel secure communication using chaotic system encoding
Energy Technology Data Exchange (ETDEWEB)
Miller, S.L.
1996-12-31
fA new method to encrypt signals using chaotic systems has been developed that offers benefits over conventional chaotic encryption methods. The method simultaneously encodes multiple plaintext streams using a chaotic system; a key is required to extract the plaintext from the chaotic cipertext. A working prototype demonstrates feasibility of the method by simultaneously encoding and decoding multiple audio signals using electrical circuits.
Combination synchronization of time-delay chaotic system via robust adaptive sliding mode control
Indian Academy of Sciences (India)
AYUB KHAN; SHIKHA
2017-06-01
In this paper, the methodology to achieve combination synchronization of time-delay chaotic system via robust adaptive sliding mode control is introduced. The methodology is implemented by taking identical time-delayLorenz chaotic system. The selection of switching surface and the design of control law is also discussed, which is an important issue. By utilizing rigorous mathematical theory, sufficient condition is drawn for the stability of error dynamics based on Lyapunov stability theory. Theoretical results are supported with the numerical simulations. The complexity of this methodology is useful to strengthen the security of communication. The hidden message can be partitioned into several parts loaded in two master systems to improve the accuracy of communication.
Combination synchronization of time-delay chaotic system via robust adaptive sliding mode control
Khan, Ayub; Shikha
2017-06-01
In this paper, the methodology to achieve combination synchronization of time-delay chaotic system via robust adaptive sliding mode control is introduced. The methodology is implemented by taking identical time-delay Lorenz chaotic system. The selection of switching surface and the design of control law is also discussed, which is an important issue. By utilizing rigorous mathematical theory, sufficient condition is drawn for the stability of error dynamics based on Lyapunov stability theory. Theoretical results are supported with the numerical simulations. The complexity of this methodology is useful to strengthen the security of communication. The hidden message can be partitioned into several parts loaded in two master systems to improve the accuracy of communication.
A Memristor-Based Complex Lorenz System and Its Modified Projective Synchronization
Directory of Open Access Journals (Sweden)
Shibing Wang
2015-11-01
Full Text Available The aim of this paper is to introduce and investigate a novel complex Lorenz system with a flux-controlled memristor, and to realize its synchronization. The system has an infinite number of stable and unstable equilibrium points, and can generate abundant dynamical behaviors with different parameters and initial conditions, such as limit cycle, torus, chaos, transient phenomena, etc., which are explored by means of time-domain waveforms, phase portraits, bifurcation diagrams, and Lyapunov exponents. Furthermore, an active controller is designed to achieve modified projective synchronization (MPS of this system based on Lyapunov stability theory. The corresponding numerical simulations agree well with the theoretical analysis, and demonstrate that the response system is asymptotically synchronized with the drive system within a short time.
Encryption in Chaotic Systems with Sinusoidal Excitations
Directory of Open Access Journals (Sweden)
G. Obregón-Pulido
2014-01-01
Full Text Available In this contribution an encryption method using a chaotic oscillator, excited by “n” sinusoidal signals, is presented. The chaotic oscillator is excited by a sum of “n” sinusoidal signals and a message. The objective is to encrypt such a message using the chaotic behavior and transmit it, and, as the chaotic system is perturbed by the sinusoidal signal, the transmission security could be increased due to the effect of such a perturbation. The procedure is based on the regulation theory and consider that the receiver knows the frequencies of the perturbing signal, with this considerations the algorithm estimates the excitation in such a way that the receiver can cancel out the perturbation and all the undesirable dynamics in order to produce only the message. In this way we consider that the security level is increased.
Modified function projective synchronization of chaotic system
Energy Technology Data Exchange (ETDEWEB)
Du Hongyue [School of Automation, Harbin University of Science and Technology, Harbin 150080 (China); Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin 150001 (China)], E-mail: du_hong_yue@yahoo.com.cn; Zeng Qingshuang; Wang Changhong [Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin 150001 (China)
2009-11-30
This paper presents a new type synchronization called modified function projective synchronization, where the drive and response systems could be synchronized up to a desired scale function matrix. It is obvious that the unpredictability of the scaling functions can additionally enhance the security of communication. By active control scheme, we take Lorenz system as an example to illustrate above synchronization phenomenon. Furthermore, based on modified function projective synchronization, a scheme for secure communication is investigated in theory. The corresponding numerical simulations are performed to verify and illustrate the analytical results.
Impulsive generalized synchronization of chaotic system
Institute of Scientific and Technical Information of China (English)
Zhang Rong; Xu Zhen-Yuan; He Xue-Ming
2007-01-01
In this paper, with a given manifold y=H(x), we have constructed a response system for a continuous-time chaotic system as a drive system, and used impulsive control theory to demonstrate theoretically that this response system can achieve impulsive generalized synchronization(GS)with the drive system. Our theoretical result is supported by numerical examples.
An adaptive strategy for controlling chaotic system
Institute of Scientific and Technical Information of China (English)
曹一家; 张红先
2003-01-01
This paper presents an adaptive strategy for controlling chaotic systems. By employing the phase space reconstruction technique in nonlinear dynamical systems theory, the proposed strategy transforms the nonlinear system into canonical form, and employs a nonlinear observer to estimate the uncertainties and disturbances of the nonlinear system, and then establishes a state-error-like feedback law. The developed control scheme allows chaos control in spite of modeling errors and parametric variations. The effectiveness of the proposed approach has been demonstrated through its applications to two well-known chaotic systems : Duffing oscillator and Rǒssler chaos.
An adaptive strategy for controlling chaotic system.
Cao, Yi-Jia; Hang, Hong-Xian
2003-01-01
This paper presents an adaptive strategy for controlling chaotic systems. By employing the phase space reconstruction technique in nonlinear dynamical systems theory, the proposed strategy transforms the nonlinear system into canonical form, and employs a nonlinear observer to estimate the uncertainties and disturbances of the nonlinear system, and then establishes a state-error-like feedback law. The developed control scheme allows chaos control in spite of modeling errors and parametric variations. The effectiveness of the proposed approach has been demonstrated through its applications to two well-known chaotic systems: Duffing oscillator and Rössler chaos.
An adaptive strategy for controlling chaotic system
Institute of Scientific and Technical Information of China (English)
曹一家; 张红先
2003-01-01
This paper presents an adaptive strategy for controlling chaotic systems. By employing the phase space reconstruction technique in nonlinear dynamical systems theory, the proposed strategy transforms the nonlinear system into canonical form, and employs a nonlinear observer to estimate the uncertainties and disturbances of the nonlinear system, and then establishes a state-error-like feedback law. The developed control scheme allows chaos control in spite of modeling errors and parametric variations. The effectiveness of the proposed approach has been demonstrated through its applications to two well-known chaotic systems: Duffing oscillator and Rossler chaos.
Synchronization of chaotic systems with different order.
Femat, Ricardo; Solís-Perales, Gualberto
2002-03-01
The chaotic synchronization of third-order systems and second-order driven oscillator is studied in this paper. Such a problem is related to synchronization of strictly different chaotic systems. We show that dynamical evolution of second-order driven oscillators can be synchronized with the canonical projection of a third-order chaotic system. In this sense, it is said that synchronization is achieved in reduced order. Duffing equation is chosen as slave system whereas Chua oscillator is defined as master system. The synchronization scheme has nonlinear feedback structure. The reduced-order synchronization is attained in a practical sense, i.e., the difference e=x(3)-x(1)(') is close to zero for all time t> or =t(0)> or =0, where t(0) denotes the time of the control activation.
Controlled transitions between cupolets of chaotic systems
Energy Technology Data Exchange (ETDEWEB)
Morena, Matthew A., E-mail: matthew.morena@wildcats.unh.edu; Short, Kevin M.; Cooke, Erica E. [Integrated Applied Mathematics Program, University of New Hampshire, Durham, New Hampshire 03824 (United States)
2014-03-15
We present an efficient control scheme that stabilizes the unstable periodic orbits of a chaotic system. The resulting orbits are known as cupolets and collectively provide an important skeleton for the dynamical system. Cupolets exhibit the interesting property that a given sequence of controls will uniquely identify a cupolet, regardless of the system's initial state. This makes it possible to transition between cupolets, and thus unstable periodic orbits, simply by switching control sequences. We demonstrate that although these transitions require minimal controls, they may also involve significant chaotic transients unless carefully controlled. As a result, we present an effective technique that relies on Dijkstra's shortest path algorithm from algebraic graph theory to minimize the transients and also to induce certainty into the control of nonlinear systems, effectively providing an efficient algorithm for the steering and targeting of chaotic systems.
Controlled transitions between cupolets of chaotic systems.
Morena, Matthew A; Short, Kevin M; Cooke, Erica E
2014-03-01
We present an efficient control scheme that stabilizes the unstable periodic orbits of a chaotic system. The resulting orbits are known as cupolets and collectively provide an important skeleton for the dynamical system. Cupolets exhibit the interesting property that a given sequence of controls will uniquely identify a cupolet, regardless of the system's initial state. This makes it possible to transition between cupolets, and thus unstable periodic orbits, simply by switching control sequences. We demonstrate that although these transitions require minimal controls, they may also involve significant chaotic transients unless carefully controlled. As a result, we present an effective technique that relies on Dijkstra's shortest path algorithm from algebraic graph theory to minimize the transients and also to induce certainty into the control of nonlinear systems, effectively providing an efficient algorithm for the steering and targeting of chaotic systems.
Generalized synchronization of two different chaotic systems
Institute of Scientific and Technical Information of China (English)
Li Guo-Hui
2007-01-01
In this paper, generalized synchronization of two different chaotic dynamical systems is investigated. An active control is adopted to construct a response system which synchronizes with a given drive system for a function relation.Based on rigorous analysis, the error system is asymptotically stable at the equilibrium. Numerical simulations illustrate the effectiveness of the proposed theory.
Urey Prize Lecture - Chaotic dynamics in the solar system
Wisdom, Jack
1987-01-01
Attention is given to solar system cases in which chaotic solutions of Newton's equations are important, as in chaotic rotation and orbital evolution. Hyperion is noted to be tumbling chaotically; chaotic orbital evolution is suggested to be of fundamental importance to an accounting for the Kirkwood gaps in asteroid distribution and for the phase space boundary of the chaotic zone at the 3/1 mean-motion commensurability with Jupiter. In addition, chaotic trajectories in the 2/1 chaotic zone reach very high eccentricities by a route that carries them to high inclinations temporarily.
A minimum principle for chaotic dynamical systems
Bracken, Paul; Góra, Paweł; Boyarsky, Abraham
2002-06-01
Discrete time dynamical systems generated by the iteration of nonlinear maps, such as the logistic map or the tent map, provide interesting examples of chaotic systems. But what is the physical principle behind the emergence of these maps? In the continuous time settings, differential equations of mechanics arise from the minimization of the energy function (Hamiltonian). However, there is no general physical principle for the discrete time analogue of differential equations, namely, maps. In this note, we present an approach to this problem. Using a natural definition of energy for chaotic systems, we minimize energy subject to the constraint that the observed dynamical system has a known entropy. We consider the case where the natural invariant measure is Lebesgue. Invoking the Euler-Lagrange equation, we derive a nonlinear second order differential equation whose solution is the chaotic map that minimizes energy.
Analysis of chaotic FM system synchronization for bistatic radar
Pappu, Chandra S.; Verdin, Berenice; Flores, Benjamin C.; Boehm, James; Debroux, Patrick
2015-05-01
We propose a scheme for bistatic radar that uses a chaotic system to generate a wideband FM signal that is reconstructed at the receiver via a conventional phase lock loop. The setup for the bistatic radar includes a 3 state variable drive oscillator at the transmitter and a response oscillator at the receiver. The challenge is in synchronizing the response oscillator of the radar receiver utilizing a scaled version of the transmitted signal sr(t, x) = αst(t, x) where x is one of three driver oscillator state variables and α is the scaling factor that accounts for antenna gain, system losses, and space propagation. For FM, we also assume that the instantaneous frequency of the received signal, xs, is a scaled version of the Lorenz variable x. Since this additional scaling factor may not be known a priori, the response oscillator must be able to accept the scaled version of x as an input. Thus, to achieve synchronization we utilize a generalized projective synchronization technique that introduces a controller term -μe where μ is a control factor and e is the difference between the response state variable xs and a scaled x. Since demodulation of sr(t) is required to reconstruct the chaotic state variable x, the phase lock loop imposes a limit on the minimum error e. We verify through simulations that, once synchronization is achieved, the short-time correlation of x and xs is high and that the self-noise in the correlation is negligible over long periods of time.
Visibility graphlet approach to chaotic time series
Energy Technology Data Exchange (ETDEWEB)
Mutua, Stephen [Business School, University of Shanghai for Science and Technology, Shanghai 200093 (China); Computer Science Department, Masinde Muliro University of Science and Technology, P.O. Box 190-50100, Kakamega (Kenya); Gu, Changgui, E-mail: gu-changgui@163.com, E-mail: hjyang@ustc.edu.cn; Yang, Huijie, E-mail: gu-changgui@163.com, E-mail: hjyang@ustc.edu.cn [Business School, University of Shanghai for Science and Technology, Shanghai 200093 (China)
2016-05-15
Many novel methods have been proposed for mapping time series into complex networks. Although some dynamical behaviors can be effectively captured by existing approaches, the preservation and tracking of the temporal behaviors of a chaotic system remains an open problem. In this work, we extended the visibility graphlet approach to investigate both discrete and continuous chaotic time series. We applied visibility graphlets to capture the reconstructed local states, so that each is treated as a node and tracked downstream to create a temporal chain link. Our empirical findings show that the approach accurately captures the dynamical properties of chaotic systems. Networks constructed from periodic dynamic phases all converge to regular networks and to unique network structures for each model in the chaotic zones. Furthermore, our results show that the characterization of chaotic and non-chaotic zones in the Lorenz system corresponds to the maximal Lyapunov exponent, thus providing a simple and straightforward way to analyze chaotic systems.
Visibility graphlet approach to chaotic time series.
Mutua, Stephen; Gu, Changgui; Yang, Huijie
2016-05-01
Many novel methods have been proposed for mapping time series into complex networks. Although some dynamical behaviors can be effectively captured by existing approaches, the preservation and tracking of the temporal behaviors of a chaotic system remains an open problem. In this work, we extended the visibility graphlet approach to investigate both discrete and continuous chaotic time series. We applied visibility graphlets to capture the reconstructed local states, so that each is treated as a node and tracked downstream to create a temporal chain link. Our empirical findings show that the approach accurately captures the dynamical properties of chaotic systems. Networks constructed from periodic dynamic phases all converge to regular networks and to unique network structures for each model in the chaotic zones. Furthermore, our results show that the characterization of chaotic and non-chaotic zones in the Lorenz system corresponds to the maximal Lyapunov exponent, thus providing a simple and straightforward way to analyze chaotic systems.
Institute of Scientific and Technical Information of China (English)
Gao Fei; Li Zhuo-Qiu; Tong Heng-Qing
2008-01-01
This paper proposes a novel quantum-behaved particle swarm optimization (NQPSO) for the estimation of chaos'unknown parameters by transforming them into nonlinear functions' optimization. By means of the techniques in the following three aspects: contracting the searching space self-adaptively; boundaries restriction strategy; substituting the particles' convex combination for their centre of mass, this paper achieves a quite effective search mechanism with fine equilibrium between exploitation and exploration. Details of applying the proposed method and other methods into Lorenz systems axe given, and experiments done show that NQPSO has better adaptability, dependability and robustness. It is a successful approach in unknown parameter estimation online especially in the cases with white noises.
Formulation of statistical mechanics for chaotic systems
Indian Academy of Sciences (India)
Vishnu M Bannur; Ramesh Babu Thayyullathil
2009-02-01
We formulate the statistical mechanics of chaotic system with few degrees of freedom and investigated the quartic oscillator system using microcanonical and canonical ensembles. Results of statistical mechanics are numerically verified by considering the dynamical evolution of quartic oscillator system with two degrees of freedom.
Blended particle filters for large-dimensional chaotic dynamical systems.
Majda, Andrew J; Qi, Di; Sapsis, Themistoklis P
2014-05-27
A major challenge in contemporary data science is the development of statistically accurate particle filters to capture non-Gaussian features in large-dimensional chaotic dynamical systems. Blended particle filters that capture non-Gaussian features in an adaptively evolving low-dimensional subspace through particles interacting with evolving Gaussian statistics on the remaining portion of phase space are introduced here. These blended particle filters are constructed in this paper through a mathematical formalism involving conditional Gaussian mixtures combined with statistically nonlinear forecast models compatible with this structure developed recently with high skill for uncertainty quantification. Stringent test cases for filtering involving the 40-dimensional Lorenz 96 model with a 5-dimensional adaptive subspace for nonlinear blended filtering in various turbulent regimes with at least nine positive Lyapunov exponents are used here. These cases demonstrate the high skill of the blended particle filter algorithms in capturing both highly non-Gaussian dynamical features as well as crucial nonlinear statistics for accurate filtering in extreme filtering regimes with sparse infrequent high-quality observations. The formalism developed here is also useful for multiscale filtering of turbulent systems and a simple application is sketched below.
Generalized Synchronization of Diverse Structure Chaotic Systems
Institute of Scientific and Technical Information of China (English)
KADIR Abdurahman; WANG Xing-Yuan; ZHAO Yu-Zhang
2011-01-01
@@ Generalized synchronization between two diverse structures of chaotic systems possesses significance in the research of synchronization.We propose an approach based on the Lyapunov stability theory to study it.This method can be used widely.Numerical examples are given to demonstrate the effectiveness of this approach.%Generalized synchronization between two diverse structures of chaotic systems possesses significance in the research of synchronization. We propose an approach based on the Lyapunov stability theory to study it. This method can be used widely. Numerical examples are given to demonstrate the effectiveness of this approach.
Generalized correlation integral vectors: A distance concept for chaotic dynamical systems
Energy Technology Data Exchange (ETDEWEB)
Haario, Heikki, E-mail: heikki.haario@lut.fi [School of Engineering Science, Lappeenranta University of Technology, Lappeenranta (Finland); Kalachev, Leonid, E-mail: KalachevL@mso.umt.edu [Department of Mathematical Sciences, University of Montana, Missoula, Montana 59812-0864 (United States); Hakkarainen, Janne [Earth Observation Unit, Finnish Meteorological Institute, Helsinki (Finland)
2015-06-15
Several concepts of fractal dimension have been developed to characterise properties of attractors of chaotic dynamical systems. Numerical approximations of them must be calculated by finite samples of simulated trajectories. In principle, the quantities should not depend on the choice of the trajectory, as long as it provides properly distributed samples of the underlying attractor. In practice, however, the trajectories are sensitive with respect to varying initial values, small changes of the model parameters, to the choice of a solver, numeric tolerances, etc. The purpose of this paper is to present a statistically sound approach to quantify this variability. We modify the concept of correlation integral to produce a vector that summarises the variability at all selected scales. The distribution of this stochastic vector can be estimated, and it provides a statistical distance concept between trajectories. Here, we demonstrate the use of the distance for the purpose of estimating model parameters of a chaotic dynamic model. The methodology is illustrated using computational examples for the Lorenz 63 and Lorenz 95 systems, together with a framework for Markov chain Monte Carlo sampling to produce posterior distributions of model parameters.
Chaotic Turing pattern formation in spatiotemporal systems
Institute of Scientific and Technical Information of China (English)
XIAO Jing-hua; LI Hai-hong; YANG Jun-zhong; HU Gang
2006-01-01
The problem of Turing pattern formation has attracted much attention in nonlinear science as well as physics,chemistry and biology.So far spatially ordered Turing patterns have been observed in stationary and oscillatory media only.In this paper we find that spatially ordered Turing patterns exist in chaotic extended systems.And chaotic Turing patterns are strikingly rich and surprisingly beautiful with their space structures.These findings are in sharp contrast with the intuition of pseudo-randomness of chaos.The richness and beauty of the chaotic Turing patterns are attributed to a large variety of symmetry properties realized by various types of self-organizations of partial chaos synchronizations.
Universal impedance fluctuations in wave chaotic systems.
Hemmady, Sameer; Zheng, Xing; Ott, Edward; Antonsen, Thomas M; Anlage, Steven M
2005-01-14
We experimentally investigate theoretical predictions of universal impedance fluctuations in wave chaotic systems using a microwave analog of a quantum chaotic infinite square well potential. We emphasize the use of the radiation impedance to remove the nonuniversal effects of the particular coupling between the outside world and the scatterer. Specific predictions that we test include the probability density functions (PDFs) of the real and imaginary parts of the universal impedance, the equality of the variances of these PDFs, and the dependence of these PDFs on a single loss parameter.
Robust synchronization of uncertain chaotic systems
Institute of Scientific and Technical Information of China (English)
Li Fang; Hu Ai-Hua; Xu Zheng-Yuan
2006-01-01
This paper investigates robust unified (lag, anticipated, and complete) synchronization of two coupled chaotic systems. By introducing the concepts of positive definite symmetrical matrix and Riccati inequality and the theory of robust stability, several criteria on robust synchronization are established. Extensive numerical simulations are also used to confirm the results.
A Design of Observers for a Discrete Chaotic System
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
It is very easy to design an observer for a discrete chaotic system which possesses one non-linear scalar quantity, and one can realize the synchronization between the investigated chaotic system and its observer easily. This method is applied to two chaotic systems.
Projective synchronization of chaotic systems with bidirectional nonlinear coupling
Indian Academy of Sciences (India)
Mohammada Ali Khan; Swarup Poria
2013-09-01
This paper presents a new scheme for constructing bidirectional nonlinear coupled chaotic systems which synchronize projectively. Conditions necessary for projective synchronization (PS) of two bidirectionally coupled chaotic systems are derived using Lyapunov stability theory. The proposed PS scheme is discussed by taking as examples the so-called unified chaotic model, the Lorenz–Stenflo system and the nonautonomous chaotic Van der Pol oscillator. Numerical simulation results are presented to show the efficiency of the proposed synchronization scheme.
Complexity and synchronization in stochastic chaotic systems
Son Dang, Thai; Palit, Sanjay Kumar; Mukherjee, Sayan; Hoang, Thang Manh; Banerjee, Santo
2016-02-01
We investigate the complexity of a hyperchaotic dynamical system perturbed by noise and various nonlinear speech and music signals. The complexity is measured by the weighted recurrence entropy of the hyperchaotic and stochastic systems. The synchronization phenomenon between two stochastic systems with complex coupling is also investigated. These criteria are tested on chaotic and perturbed systems by mean conditional recurrence and normalized synchronization error. Numerical results including surface plots, normalized synchronization errors, complexity variations etc show the effectiveness of the proposed analysis.
SWITCHING CONTROL:FROM SIMPLE RULES TO COMPLEX CHAOTIC SYSTEMS
Institute of Scientific and Technical Information of China (English)
L(U) Jinhu
2003-01-01
This paper reviews and introduces some simple switching piecewise-linear controllers, which can generate complex chaotic behaviors from simple switching systems. The mechanism of simple switching rules creating complex chaotic behaviors is further investigated.
CHAOTIC CONTROL OF NONLINEAR SYSTEMS BASED ON IMPROVED CORRELATIVITY
Institute of Scientific and Technical Information of China (English)
ZhouZiaoan; ZhangJihong
2003-01-01
Chaotic sequences are basically ergodic random esquences.By improving correlativity of a chaotic signal,the chaotic dynamic system can be controlled to converge to its equilibrium point and,more significantly,to its multi-periodic orbits.Mathematical theory analysis is carried out and some computer simulation results are provided to support such controllability of the chaotic Henon system and the discrete coupled map lattice.
Chaotic Phenomena in Technical Control Systems
DEFF Research Database (Denmark)
Mosekilde, Erik
1997-01-01
The paper discusses a number of examples of technical control systems that can exhibit deterministic chaos and other forms of complex nonlinear behavior. These examples include thermostatically regulated radiators, closely placed refrigirators, and industrial cooling compressors. The paper...... continues to describe the possible perspective in driving our technical systems to operate in a chaotic regime. An example of a technical system capable of operating under unstable conditions is the F/A-18 fighter....
A Parameter Modulation Chaotic Secure Communication Scheme with Channel Noises
Institute of Scientific and Technical Information of China (English)
ZHANG Ying-Qian; WANG Xing-Yuan
2011-01-01
@@ We propose a new communication system which is able to separate noise successfully by using independent component analysis (ICA), and a parameter modulation method based on a Lorenz chaotic system is employed for recovery of the source signals.The results indicate that our proposed secure communication has robustness against noise.%We propose a new communication system which is able to separate noise successfully by using independent component analysis (ICA), and a parameter modulation method based on a Lorenz chaotic system is employed for recovery of the source signals. The results indicate that our proposed secure communication has robustness against noise.
Indian Academy of Sciences (India)
Fei Yu; Chunhua Wang; Qiuzhen Wan; Yan Hu
2013-02-01
A five-term three-dimensional (3D) autonomous chaotic system with an exponential nonlinear term is reported in this paper. Basic dynamical behaviours of the chaotic system are further investigated. Then a new synchronization phenomenon, complete switched modified function projective synchronization (CSMFPS), for this novel five-term chaotic system with uncertain parameters and disturbances is investigated. This paper extends previous work, where CSMFPS of chaotic systems means that all the state variables of the drive system synchronize with different state variables of the response system. As the synchronization scheme has many combined forms, it is a promising type of synchronization and can provide greater security in secure communication. Based on Lyapunov stability theory, a robust adaptive controller is contrived to acquire CSMFPS, parameter identification and suppress disturbances simultaneously. Finally, the Lorenz system and the new five-term chaotic system are taken as examples and the corresponding numerical simulations are presented to verify the effectiveness and feasibility of the proposed control scheme.
Cascade adaptive control of uncertain unified chaotic systems
Institute of Scientific and Technical Information of China (English)
Wei Wei; Li Dong-Hai; Wang Jing
2011-01-01
The chaos control of uncertain unified chaotic systems is considered. Cascade adaptive control approach with only one control input is presented to stabilize states of the uncertain unified chaotic system at the zero equilibrium point.Since an adaptive controller based on dynamic compensation mechanism is employed, the exact model of the unified chaotic system is not necessarily required.By choosing appropriate controller parameters, chaotic phenomenon can be suppressed and the response speed is tunable. Sufficient condition for the asymptotic stability of the approach is derived. Numerical simulation results confirm that the cascade adaptive control approach with only one control signal is valid in chaos control of uncertain unified chaotic systems.
Quantum chaotic attractor in a dissipative system
Liu, W V; Schieve, William C.
1997-01-01
A dissipative quantum system is treated here by coupling it with a heat bath of harmonic oscillators. Through quantum Langevin equations and Ehrenfest's theorem, we establish explicitly the quantum Duffing equations with a double-well potential chosen. A quantum noise term appears the only driving force in dynamics. Numerical studies show that the chaotic attractor exists in this system while chaos is certainly forbidden in the classical counterpart.
Nonuniversality of weak synchronization in chaotic systems
Vieira, M. de Sousa; Lichtenberg, A.J.
1997-01-01
We show that the separate properties of weak synchronization (WS) and strong synchronization (SS), reported recently by Pyragas [K. Pyragas, Phys. Rev. E, 54, R4508 (1996)], in unidirectionally coupled chaotic systems, are not generally distinct properties of such systems. In particular, we find analytically for the tent map and numerically for some parameters of the circle map that the transition to WS and SS coincide.
Frequency-Locking in Coupled Chaotic Systems
Institute of Scientific and Technical Information of China (English)
HU Bam-Bi; LIU Zong-Hua; ZHENG Zhi-Gang
2001-01-01
A novel approach is presented for measuring the phase synchronization (frequency-locking) of coupled N nonidentical oscillators, which can characterize frequency-locking for chaotic systems without well-defined phase by measuring the mean frequency. Numerical simulations confirm the existence of frequency-locking. The relations between the mean frequency and the coupling strength and the frequency mismatch are given. For the coupled hyperchaotic systems, the frequency-locking can be better characterized by more than one mean frequency curves.
Entanglement production in Quantized Chaotic Systems
Bandyopadhyay, J N; Bandyopadhyay, Jayendra N.; Lakshminarayan, Arul
2005-01-01
Quantum chaos is a subject whose major goal is to identify and to investigate different quantum signatures of classical chaos. Here we study entanglement production in coupled chaotic systems as a possible quantum indicator of classical chaos. We use coupled kicked tops as a model for our extensive numerical studies. We find that, in general, presence of chaos in the system produces more entanglement. However, coupling strength between two subsystems is also very important parameter for the entanglement production. Here we show how chaos can lead to large entanglement which is universal and describable by random matrix theory (RMT). We also explain entanglement production in coupled strongly chaotic systems by deriving a formula based on RMT. This formula is valid for arbitrary coupling strengths, as well as for sufficiently long time. Here we investigate also the effect of chaos on the entanglement production for the mixed initial state. We find that many properties of the mixed state entanglement production...
Henon CSK Secure Communication System Using Chaotic Turbo Codes
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
In this paper,the authors design a novel chaotic secure communication system, which has high security and good errorcorrecting capability. Firstly, the Henon Chaos Shift Keying (CSK) modulation block is presented. Secondly,chaotic turbo encod er/decoder (hard decision) is introduced. Thirdly, this chaotic secure communication system, which comprises the Henon CSK modulation block and chaotic turbo en coder in a serially concatenated form, is shown. Furthermore, a novel two step encryption scheme is proposed, which is based on the chaotic turbo e ncoded Henon CSK secure communication system.
New robust chaotic system with exponential quadratic term
Institute of Scientific and Technical Information of China (English)
Bao Bo-Cheng; Li Chun-Biao; Xu Jian-Peing; Liu Zhong
2008-01-01
This paper proposes a new robust chaotic system of three-dimensional quadratic autonomous ordinary differential equations by introducing an exponential quadratic term.This system can display a double-scroll chaotic attractor with only two equilibria,and can be found to be robust chaotic in a very wide parameter domain with positive maximum Lyapunov exponent.Some basic dynamical properties and chaotic behaviour of novel attractor are studied.By numerical simulation,this paper verifies that the three-dimensional system can also evolve into periodic and chaotic behaviottrs by a constant controller.
Institute of Scientific and Technical Information of China (English)
Wang Xing-Yuan; Bao Xue-Mei
2013-01-01
In this paper,we propose a novel block cryptographic scheme based on a spatiotemporal chaotic system and a chaotic neural network (CNN).The employed CNN comprises a 4-neuron layer called a chaotic neuron layer (CNL),where the spatiotemporal chaotic system participates in generating its weight matrix and other parameters.The spatiotemporal chaotic system used in our scheme is the typical coupled map lattice (CML),which can be easily implemented in parallel by hardware.A 160-bit-long binary sequence is used to generate the initial conditions of the CML.The decryption process is symmetric relative to the encryption process.Theoretical analysis and experimental results prove that the block cryptosystem is secure and practical,and suitable for image encryption.
Entanglement production in quantized chaotic systems
Indian Academy of Sciences (India)
Jayendra N Bandyopadhyay; Arul Lakshminarayan
2005-04-01
Quantum chaos is a subject whose major goal is to identify and to investigate different quantum signatures of classical chaos. Here we study entanglement production in coupled chaotic systems as a possible quantum indicator of classical chaos. We use coupled kicked tops as a model for our extensive numerical studies. We find that, in general, chaos in the system produces more entanglement. However, coupling strength between two subsystems is also a very important parameter for entanglement production. Here we show how chaos can lead to large entanglement which is universal and describable by random matrix theory (RMT). We also explain entanglement production in coupled strongly chaotic systems by deriving a formula based on RMT. This formula is valid for arbitrary coupling strengths, as well as for sufficiently long time. Here we investigate also the effect of chaos on the entanglement production for the mixed initial state. We find that many properties of the mixed-state entanglement production are qualitatively similar to the pure state entanglement production. We however still lack an analytical understanding of the mixed-state entanglement production in chaotic systems.
Exact Eigenfunctions of a Chaotic System
Ausländer, O M
1997-01-01
The interest in the properties of quantum systems, whose classical dynamics are chaotic, derives from their abundance in nature. The spectrum of such systems can be related, in the semiclassical approximation (SCA), to the unstable classical periodic orbits, through Gutzwiller's trace formula. The class of systems studied in this work, tiling billiards on the pseudo-sphere, is special in this correspondence being exact, via Selberg's trace formula. In this work, an exact expression for Green's function (GF) and the eigenfunctions (EF) of tiling billiards on the pseudo-sphere, whose classical dynamics are chaotic, is derived. GF is shown to be equal to the quotient of two infinite sums over periodic orbits, where the denominator is the spectral determinant. Such a result is known to be true for typical chaotic systems, in the leading SCA. From the exact expression for GF, individual EF can be identified. In order to obtain a SCA by finite series for the infinite sums encountered, resummation by analytic contin...
Resonance eigenfunctions in chaotic scattering systems
Indian Academy of Sciences (India)
Martin Sieber
2009-09-01
We study the semiclassical structure of resonance eigenstates of open chaotic systems. We obtain semiclassical estimates for the weight of these states on different regions in phase space. These results imply that the long-lived right (left) eigenstates of the non-unitary propagator are concentrated in the semiclassical limit ħ → 0 on the backward (forward) trapped set of the classical dynamics. On this support the eigenstates display a self-similar behaviour which depends on the limiting decay rate.
On fuzzy sampled-data control of chaotic systems via a time-dependent Lyapunov functional approach.
Wang, Zi-Peng; Wu, Huai-Ning
2015-04-01
In this paper, a novel approach to fuzzy sampled-data control of chaotic systems is presented by using a time-dependent Lyapunov functional. The advantage of the new method is that the Lyapunov functional is continuous at sampling times but not necessarily positive definite inside the sampling intervals. Compared with the existing works, the constructed Lyapunov functional makes full use of the information on the piecewise constant input and the actual sampling pattern. In terms of a new parameterized linear matrix inequality (LMI) technique, a less conservative stabilization condition is derived to guarantee the exponential stability for the closed-loop fuzzy sampled-data system. By solving a set of LMIs, the fuzzy sampled-data controller can be easily obtained. Finally, the chaotic Lorenz system and Rössler's system are employed to illustrate the feasibility and effectiveness of the proposed method.
Improving the prediction of chaotic time series
Institute of Scientific and Technical Information of China (English)
李克平; 高自友; 陈天仑
2003-01-01
One of the features of deterministic chaos is sensitive to initial conditions. This feature limits the prediction horizons of many chaotic systems. In this paper, we propose a new prediction technique for chaotic time series. In our method, some neighbouring points of the predicted point, for which the corresponding local Lyapunov exponent is particularly large, would be discarded during estimating the local dynamics, and thus the error accumulated by the prediction algorithm is reduced. The model is tested for the convection amplitude of Lorenz systems. The simulation results indicate that the prediction technique can improve the prediction of chaotic time series.
Robust lag synchronization between two different chaotic systems via dual-stage impulsive control
Institute of Scientific and Technical Information of China (English)
Zhang Hua-Guang; Ma Tie-Dong; Fu Jie; Tong Shao-Cheng
2009-01-01
In this paper,an improved impulsive lag synchronization scheme for different chaotic systems with parametric uncertainties is proposed. Based on the new definition of synchronization with error bound and a novel impulsive control scheme (the so-called duai-stage impulsive control),some new and less conservative sufficient conditions are established to guarantee that the error dynamics can converge to a predetermined level,which is more reasonable and rigorous than the existing results. In particular,some simpler and more convenient conditions are derived by taking the same impulsive distances and control gains. Finally,some numerical simulations for the Lorenz system and the Chen system are given to demonstrate the effectiveness and feasibility of the proposed method.
Localization of compact invariant sets of the Lorenz' 1984 model
Starkov, K. E.
In 1984 E. Lorenz published a paper [1] in which he proposed "the simplest possible general circulation model": dot{x} = -y^2 - z^2 - ax + aF, dot{y} = xy -bxz - y+G, dot{z} = bxy + xz -z which is referred to as the Lorenz'1984 model. The existence of chaos was shown in [1, 2] for different values of parameters. Dynamical studies of this system were realized in papers [1, 2]; [3], [4]. This paper is devoted to study of a localization problem of compact invariant sets of the Lorenz'1984 model with help of one approach elaborated in papers of Krishchenko and Starkov, see e.g. [5]. This problem is an important topic in studies of dynamics of a chaotic system because of the interest to a long-time behavior of a system. In this work we establish that all compact invariant sets of the Lorenz' 1984 model are contained in the set \\{ x le F;x^2 + y^2 + z^2 le η ^2 = {2left( {a + 2} right)F^2 + 3G^2 + 2Gsqrt {aF^2 + G^2 } }/4\\} . Further, we improve this localization with help of refining bound η using additional localizations sets. By applying cylindrical coordinates to the Lorenz' 1984 model we derive yet another localization set of the form \\{ y^2 + z^2 le G^2 (1 + b^{ - 2} )exp (4π b^{ - 1} )\\}. Finally, we discuss how to improve the final localization set and consider one example.
Fractional order control and synchronization of chaotic systems
Vaidyanathan, Sundarapandian; Ouannas, Adel
2017-01-01
The book reports on the latest advances in and applications of fractional order control and synchronization of chaotic systems, explaining the concepts involved in a clear, matter-of-fact style. It consists of 30 original contributions written by eminent scientists and active researchers in the field that address theories, methods and applications in a number of research areas related to fractional order control and synchronization of chaotic systems, such as: fractional chaotic systems, hyperchaotic systems, complex systems, fractional order discrete chaotic systems, chaos control, chaos synchronization, jerk circuits, fractional chaotic systems with hidden attractors, neural network, fuzzy logic controllers, behavioral modeling, robust and adaptive control, sliding mode control, different types of synchronization, circuit realization of chaotic systems, etc. In addition to providing readers extensive information on chaos fundamentals, fractional calculus, fractional differential equations, fractional contro...
ADAPTIVE CONTROL AND IDENTIFICATION OF CHAOTIC SYSTEMS
Institute of Scientific and Technical Information of China (English)
LI ZHI; HAN CHONG-ZHAO
2001-01-01
A novel adaptive control and identification on-line method is proposed for a class of chaotic system with uncertain parameters. We prove that, using the presented method, a controller and identifier is developed which can remove chaos in nonlinear systems and make the system asymptotically stabilizing to an arbitrarily desired smooth orbit. And at the same time, estimates to uncertain parameters converge to their true values. The advantage of our method over the existing result is that the controller and identifier is directly constructed by analytic formula without knowing unknown bounds about uncertain parameters in advance. A computer simulation example is given to validate the proposed approach.
Universal and nonuniversal properties of wave-chaotic scattering systems.
Yeh, Jen-Hao; Hart, James A; Bradshaw, Elliott; Antonsen, Thomas M; Ott, Edward; Anlage, Steven M
2010-02-01
Prediction of the statistics of scattering in typical wave-chaotic systems requires combining system-specific information with universal aspects of chaotic scattering as described by random matrix theory. This Rapid Communication shows that the average impedance matrix, which characterizes such system-specific properties, can be semiclassically calculated in terms of ray trajectories between ports. Theoretical predictions are compared with experimental results for a microwave billiard, demonstrating that the theory successfully uncovered universal statistics of wave-chaotic scattering systems.
Aires, Filipe; Rossow, William B.; Hansen, James E. (Technical Monitor)
2001-01-01
A new approach is presented for the analysis of feedback processes in a nonlinear dynamical system by observing its variations. The new methodology consists of statistical estimates of the sensitivities between all pairs of variables in the system based on a neural network modeling of the dynamical system. The model can then be used to estimate the instantaneous, multivariate and nonlinear sensitivities, which are shown to be essential for the analysis of the feedbacks processes involved in the dynamical system. The method is described and tested on synthetic data from the low-order Lorenz circulation model where the correct sensitivities can be evaluated analytically.
ON FEEDBACK CONTROL OF DELAYED CHAOTIC SYSTEM
Institute of Scientific and Technical Information of China (English)
李丽香; 彭海朋; 卢辉斌; 关新平
2001-01-01
In this paper two different types of feedback control technique are discussed: the standard feedback control and the time-delay feedback control which have been successfully used in many control systems. In order to understand to what extent the two different types of control technique are useful in delayed chaotic systems, some analytic stabilization conditions for chaos control from the two types of control technique are derived based on Lyapunov stabilization arguments. Similarly, we discuss the tracking problem by applying the time-delay feedback control. Finally, numerical examples are provided.
Control of a Unified Chaotic System via Single Variable Feedback
Guo, Rong-Wei; Vincent E., U.
2009-09-01
Based on the LaSalle invariance principle, we propose a simple adaptive-feedback for controlling the unified chaotic system. We show explicitly with numerical proofs that our method can easily achieve the control of chaos in the unified chaotic system using only a single variable feedback. The present controller, to our knowledge, is the simplest control scheme for controlling a unified chaotic system.
Circuit realization of the fractional-order unified chaotic system
Institute of Scientific and Technical Information of China (English)
Chen Xiang-Rong; Liu Chong-Xin; Wang Fa-Qiang
2008-01-01
This paper studies the chaotic behaviours of the fractional-order unified chaotic system.Based on the approximation method in frequency domain,it proposes an electronic circuit model of tree shape to realize the fractional-order operator.According to the tree shape model,an electronic circuit is designed to realize the 2.7-order unified chaotic system.Numerical simulations and circuit experiments have verified the existence of chaos in the fraction-order unified system.
Projective Synchronization in Time-Delayed Chaotic Systems
Institute of Scientific and Technical Information of China (English)
FENG Cun-Fang; ZHANG Yan; WANG Ying-Hai
2006-01-01
For the first time, we report on projective synchronization between two time delay chaotic systems with single time delays. It overcomes some limitations of the previous wort, where projective synchronization has been investigated only in finite-dimensional chaotic systems, so we can achieve projective synchronization in infinite-dimensional chaotic systems. We give a general method with which we can achieve projective synchronization in time-delayed chaotic systems. The method is illustrated using the famous delay-differential equations related to optical bistability. Numerical simulations fully support the analytical approach.
Adaptive generalized functional synchronization of Chaotic systems with unknown parameters
Institute of Scientific and Technical Information of China (English)
Wang Dong-Feng; Han Pu
2008-01-01
A universal adaptive generalized functional synchronization approach to any two different or identical chaotic systems with unknown parameters is proposed,based on a unified mathematical expression of a large class of chaotic system.Self-adaptive parameter law and control law are given in the form of a theorem.The synchronization between the three-dimensional R(o)ssler chaotic system and the four-dimensional Chen's hyper-chaotic system is studied as an example for illustration.The computer simulation results demonstrate the feasibility of the method proposed.
Institute of Scientific and Technical Information of China (English)
WANG Jian-Gen; ZHAO Yi
2005-01-01
@@ We propose a Bang-Bang control scheme that can synchronize master-slave chaotic systems. The chaotic systems considered here are structurally different from each other. Different from some control strategies reported previously, the scheme proposed here can be taken as a generalone that is independent of the chaotic system itself.
Modified scaling function projective synchronization of chaotic systems
Xu, Yu-Hua; Zhou, Wu-Neng; Fang, Jian-An
2011-09-01
This paper investigates a kind of modified scaling function projective synchronization of uncertain chaotic systems using an adaptive controller. The given scaling function in the new method can be an equilibrium point, a periodic orbit, or even a chaotic attractor in the phase space. Based on LaSalle's invariance set principle, the adaptive control law is derived to make the states of two chaotic systems function projective synchronized. Some numerical examples are also given to show the effectiveness of the proposed method.
Implementation of a new chaotic encryption system and synchronization
Institute of Scientific and Technical Information of China (English)
Long Min; Peng Fei; Qiu Shuisheng; Chen Yanfeng
2006-01-01
A new kind of secure communication system which combines the chaotic encryption means with the conventional encryption method is discussed. With the analysis results and the experiment data, the anti-attack ability of this communication system is significantly improved compared to that of the either method. At the same time, a new method of chaotic synchronization is proposed. With a small mixed discrete chaotic signal, it is quickly to synchronize the communication and a good security performance is ensured.
Projective synchronization in fractional order chaotic systems and its control
Li, Chunguang
2006-01-01
The chaotic dynamics of fractional (non-integer) order systems have begun to attract much attention in recent years. In this paper, we study the projective synchronization in two coupled fractional order chaotic oscillators. It is shown that projective synchronization can also exist in coupled fractional order chaotic systems. A simple feedback control method for controlling the scaling factor onto a desired value is also presented.
Noise-Induced Riddling in Chaotic Systems
Energy Technology Data Exchange (ETDEWEB)
Lai, Y.; Grebogi, C. [Departments of Physics and Astronomy and of Mathematics, The University of Kansas, Lawrence, Kansas 66045 (United States)]|[Institute for Plasma Research, The University of Maryland, College Park, Maryland 20742 (United States)
1996-12-01
Recent works have considered the situation of riddling where, when a chaotic attractor lying in an invariant subspace is {ital transversely} {ital stable}, the basin of the attractor can be riddled with holes that belong to the basin of another attractor. We show that riddling can be induced by arbitrarily small random noise {ital even} {ital if} {ital the} {ital attractor} {ital is} {ital transversely} {ital unstable}, and we obtain universal scaling laws for noise-induced riddling. Our results imply that the phenomenon of riddling can be more prevalent than expected before, as noise is practically inevitable in dynamical systems. {copyright} {ital 1996 The American Physical Society.}
Quantum ratchets in dissipative chaotic systems.
Carlo, Gabriel G; Benenti, Giuliano; Casati, Giulio; Shepelyansky, Dima L
2005-04-29
Using the method of quantum trajectories, we study a quantum chaotic dissipative ratchet appearing for particles in a pulsed asymmetric potential in the presence of a dissipative environment. The system is characterized by directed transport emerging from a quantum strange attractor. This model exhibits, in the limit of small effective Planck constant, a transition from quantum to classical behavior, in agreement with the correspondence principle. We also discuss parameter values suitable for the implementation of the quantum ratchet effect with cold atoms in optical lattices.
Infinite-Dimensional Linear Dynamical Systems with Chaoticity
Fu Xin Chu; Fu, Xin-Chu; Duan, Jinqiao
1998-01-01
The authors present two results on infinite-dimensional linear dynamical systems with chaoticity. One is about the chaoticity of the backward shift map in the space of infinite sequences on a general Fréchet space. The other is about the chaoticity of a translation map in the space of real continuous functions. The chaos is shown in the senses of both Li-Yorke and Wiggins. Treating dimensions as freedoms, the two results imply that in the case of an infinite number of freedoms, a system may exhibit complexity even when the action is linear. Finally, the authors discuss physical applications of infinite-dimensional linear chaotic dynamical systems.
Towards generalized synchronization of strictly different chaotic systems
Energy Technology Data Exchange (ETDEWEB)
Femat, R. [Matematicas Aplicadas y Sistemas Computacionales, IPICYT, Apdo. Postal 3-90, 78291 Tangamanga, San Luis Potosi S.L.P. (Mexico)]. E-mail: rfemat@ipicyt.edu.mx; Kocarev, L. [Institute for Nonlinear Science, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0402 (United States)]. E-mail: lkocarev@ucsd.edu; Gerven, L. van [Department of Mechanical Engineering, Technische Universiteit Eindhoven (Netherlands); Monsivais-Perez, M.E. [Matematicas Aplicadas y Sistemas Computacionales, IPICYT, Camino a la Presa San Jose 2055, 78216 Lomas 4a Sec., San Luis Potosi S.L.P. (Mexico)
2005-07-11
This contribution addresses the problem of the generalized synchronization (GS) in different chaotic systems, and departs from chaotic systems in a triangular from, which can be derived from Lie derivatives. A state-feedback (full knowledge of both master and slave systems) scheme is designed, which achieves GS. The work includes illustrative examples; moreover an experimental setup is used to corroborate the obtained results.
Adaptive Synchronization of Memristor-based Chaotic Neural Systems
Directory of Open Access Journals (Sweden)
Xiaofang Hu
2014-11-01
Full Text Available Chaotic neural networks consisting of a great number of chaotic neurons are able to reproduce the rich dynamics observed in biological nervous systems. In recent years, the memristor has attracted much interest in the efficient implementation of artificial synapses and neurons. This work addresses adaptive synchronization of a class of memristor-based neural chaotic systems using a novel adaptive backstepping approach. A systematic design procedure is presented. Simulation results have demonstrated the effectiveness of the proposed adaptive synchronization method and its potential in practical application of memristive chaotic oscillators in secure communication.
Impulsive Control for Fractional-Order Chaotic Systems
Institute of Scientific and Technical Information of China (English)
ZHONG Qi-Shui; BAO Jing-Fu; YU Yong-Bin; LIAO Xiao-Feng
2008-01-01
@@ We propose an impulsive control scheme for fractional-order chaotic systems. Based on the Takagi-Sugeno (T-S) fuzzy model and linear matrix inequalities (LMIs), some sufficient conditions are given to stabilize the fractional-order chaotic system via impulsive control. Numerical simulation shows the effectiveness of this approach.
Persistent excitation in adaptive parameter identification of uncertain chaotic system
Institute of Scientific and Technical Information of China (English)
Zhao Jun-chan; Zhang Qun-Jiao; Lu Jun-An
2011-01-01
This paper studies the parameter identification problem of chaotic systems. Adaptive identification laws are proposed to estimate the parameters of uncertain chaotic systems. It proves that the asymptotical identification is ensured by a persistently exciting condition. Additionally, the method can be applied to identify the uncertain parameters with any number. Numerical simulations are given to validate the theoretical analysis.
Chaotic Behavior in a Switched Dynamical System
Directory of Open Access Journals (Sweden)
Fatima El Guezar
2008-01-01
Full Text Available We present a numerical study of an example of piecewise linear systems that constitute a class of hybrid systems. Precisely, we study the chaotic dynamics of the voltage-mode controlled buck converter circuit in an open loop. By considering the voltage input as a bifurcation parameter, we observe that the obtained simulations show that the buck converter is prone to have subharmonic behavior and chaos. We also present the corresponding bifurcation diagram. Our modeling techniques are based on the new French native modeler and simulator for hybrid systems called Scicos (Scilab connected object simulator which is a Scilab (scientific laboratory package. The followed approach takes into account the hybrid nature of the circuit.
Cryptography based on spatial chaotic system
Sun, Fuyan; Lü, Zongwang
2010-08-01
Encryption of images is different from that of texts due to some intrinsic features of images such as bulk data capacity and high redundancy, which is generally difficult to handle by traditional methods. This paper proposes a new spatial chaos system(SCS), which is investigated by conducting FIPS 140-1 statistic test, and is especially useful for encryption of digital images. It is shown how to adapt a two dimensional(2D) ergodic matrix obtained from SCS to permute the positions of image pixels and confuse the relationship between the cipher image and plain image simultaneously. Experimental results show that the performance and security of the proposed cryptographic system are better than those of existing lower dimensional chaotic cryptographic systems.
Chaotic Disintegration of the Inner Solar System
Batygin, Konstantin; Holman, Mathew J
2014-01-01
On timescales that greatly exceed an orbital period, typical planetary orbits evolve in a stochastic yet stable fashion. On even longer timescales, however, planetary orbits can spontaneously transition from bounded to unbound chaotic states. Large-scale instabilities associated with such behavior appear to play a dominant role in shaping the architectures of planetary systems, including our own. Here we show how such transitions are possible, focusing on the specific case of the long-term evolution of Mercury. We develop a simple analytical model for Mercury's dynamics and elucidate the origins of its short term stochastic behavior as well as of its sudden progression to unbounded chaos. Our model allows us to estimate the timescale on which this transition is likely to be triggered, i.e. the dynamical lifetime of the Solar System as we know it. The formulated theory is consistent with the results of numerical simulations and is broadly applicable to extrasolar planetary systems dominated by secular interact...
Nonlinear Dynamics, Chaotic and Complex Systems
Infeld, E.; Zelazny, R.; Galkowski, A.
2011-04-01
Part I. Dynamic Systems Bifurcation Theory and Chaos: 1. Chaos in random dynamical systems V. M. Gunldach; 2. Controlling chaos using embedded unstable periodic orbits: the problem of optimal periodic orbits B. R. Hunt and E. Ott; 3. Chaotic tracer dynamics in open hydrodynamical flows G. Karolyi, A. Pentek, T. Tel and Z. Toroczkai; 4. Homoclinic chaos L. P. Shilnikov; Part II. Spatially Extended Systems: 5. Hydrodynamics of relativistic probability flows I. Bialynicki-Birula; 6. Waves in ionic reaction-diffusion-migration systems P. Hasal, V. Nevoral, I. Schreiber, H. Sevcikova, D. Snita, and M. Marek; 7. Anomalous scaling in turbulence: a field theoretical approach V. Lvov and I. Procaccia; 8. Abelian sandpile cellular automata M. Markosova; 9. Transport in an incompletely chaotic magnetic field F. Spineanu; Part III. Dynamical Chaos Quantum Physics and Foundations Of Statistical Mechanics: 10. Non-equilibrium statistical mechanics and ergodic theory L. A. Bunimovich; 11. Pseudochaos in statistical physics B. Chirikov; 12. Foundations of non-equilibrium statistical mechanics J. P. Dougherty; 13. Thermomechanical particle simulations W. G. Hoover, H. A. Posch, C. H. Dellago, O. Kum, C. G. Hoover, A. J. De Groot and B. L. Holian; 14. Quantum dynamics on a Markov background and irreversibility B. Pavlov; 15. Time chaos and the laws of nature I. Prigogine and D. J. Driebe; 16. Evolutionary Q and cognitive systems: dynamic entropies and predictability of evolutionary processes W. Ebeling; 17. Spatiotemporal chaos information processing in neural networks H. Szu; 18. Phase transitions and learning in neural networks C. Van den Broeck; 19. Synthesis of chaos A. Vanecek and S. Celikovsky; 20. Computational complexity of continuous problems H. Wozniakowski; Part IV. Complex Systems As An Interface Between Natural Sciences and Environmental Social and Economic Sciences: 21. Stochastic differential geometry in finance studies V. G. Makhankov; Part V. Conference Banquet
Chiral scars in chaotic Dirac fermion systems.
Xu, Hongya; Huang, Liang; Lai, Ying-Cheng; Grebogi, Celso
2013-02-08
Do relativistic quantum scars in classically chaotic systems possess unique features that are not shared by nonrelativistic quantum scars? We report a class of relativistic quantum scars in massless Dirac fermion systems whose phases return to the original values or acquire a 2π change only after circulating twice about some classical unstable periodic orbits. We name such scars chiral scars, the successful identification of which has been facilitated tremendously by our development of an analytic, conformal-mapping-based method to calculate an unprecedentedly large number of eigenstates with high accuracy. Our semiclassical theory indicates that the physical origin of chiral scars can be attributed to a combined effect of chirality intrinsic to massless Dirac fermions and the geometry of the underlying classical orbit.
Chaotic Diffusion in the Gliese-876 Planetary System
Martí, J G; Beaugé, C
2016-01-01
Chaotic diffusion is supposed to be responsible for orbital instabilities in planetary systems after the dissipation of the protoplanetary disk, and a natural consequence of irregular motion. In this paper we show that resonant multi-planetary systems, despite being highly chaotic, not necessarily exhibit significant diffusion in phase space, and may still survive virtually unchanged over timescales comparable to their age.Using the GJ-876 system as an example, we analyze the chaotic diffusion of the outermost (and less massive) planet. We construct a set of stability maps in the surrounding regions of the Laplace resonance. We numerically integrate ensembles of close initial conditions, compute Poincar\\'e maps and estimate the chaotic diffusion present in this system. Our results show that, the Laplace resonance contains two different regions: an inner domain characterized by low chaoticity and slow diffusion, and an outer one displaying larger values of dynamical indicators. In the outer resonant domain, th...
Chaotic dynamics of a Chua's system with voltage controllability
Heo, Yun Seok; Jung, Jin Woo; Kim, Ji Man; Jo, Mun Kyu; Song, Han Jung
2012-04-01
This paper presents an integrated circuit oriented Chua's chaotic system with voltage controllability. The proposed chaotic system consists of an OTA (Operational Transconductance Amplifier)-based ground inductor, two passive capacitors, a MOS (Metal-Oxide-Semiconductor)-based active resistor and an OTA-based Chua's diode with negative nonlinearity. A SPICE (Simulation Program with Integrated Circuit Emphasis) circuit analysis using 0.5-µm CMOS (Complementary Metal-Oxide-Semiconductor) process parameters was performed for the chaotic dynamics, such as the time waveform and the attractor plot. We confirmed that the chaotic behaviors of the system could be controlled by using the gate voltage of the MOS-based active resistor. Also, various chaotic dynamics of the circuit were analyzed for various MOS sizes of the OTA in the Chua's diode.
Energy Technology Data Exchange (ETDEWEB)
Sheng, Zheng, E-mail: 19994035@sina.com [College of Meteorology and Oceanography, PLA University of Science and Technology, Nanjing 211101 (China); Wang, Jun; Zhou, Bihua [National Defense Key Laboratory on Lightning Protection and Electromagnetic Camouflage, PLA University of Science and Technology, Nanjing 210007 (China); Zhou, Shudao [College of Meteorology and Oceanography, PLA University of Science and Technology, Nanjing 211101 (China); Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters, Nanjing University of Information Science and Technology, Nanjing 210044 (China)
2014-03-15
This paper introduces a novel hybrid optimization algorithm to establish the parameters of chaotic systems. In order to deal with the weaknesses of the traditional cuckoo search algorithm, the proposed adaptive cuckoo search with simulated annealing algorithm is presented, which incorporates the adaptive parameters adjusting operation and the simulated annealing operation in the cuckoo search algorithm. Normally, the parameters of the cuckoo search algorithm are kept constant that may result in decreasing the efficiency of the algorithm. For the purpose of balancing and enhancing the accuracy and convergence rate of the cuckoo search algorithm, the adaptive operation is presented to tune the parameters properly. Besides, the local search capability of cuckoo search algorithm is relatively weak that may decrease the quality of optimization. So the simulated annealing operation is merged into the cuckoo search algorithm to enhance the local search ability and improve the accuracy and reliability of the results. The functionality of the proposed hybrid algorithm is investigated through the Lorenz chaotic system under the noiseless and noise condition, respectively. The numerical results demonstrate that the method can estimate parameters efficiently and accurately in the noiseless and noise condition. Finally, the results are compared with the traditional cuckoo search algorithm, genetic algorithm, and particle swarm optimization algorithm. Simulation results demonstrate the effectiveness and superior performance of the proposed algorithm.
Sheng, Zheng; Wang, Jun; Zhou, Shudao; Zhou, Bihua
2014-03-01
This paper introduces a novel hybrid optimization algorithm to establish the parameters of chaotic systems. In order to deal with the weaknesses of the traditional cuckoo search algorithm, the proposed adaptive cuckoo search with simulated annealing algorithm is presented, which incorporates the adaptive parameters adjusting operation and the simulated annealing operation in the cuckoo search algorithm. Normally, the parameters of the cuckoo search algorithm are kept constant that may result in decreasing the efficiency of the algorithm. For the purpose of balancing and enhancing the accuracy and convergence rate of the cuckoo search algorithm, the adaptive operation is presented to tune the parameters properly. Besides, the local search capability of cuckoo search algorithm is relatively weak that may decrease the quality of optimization. So the simulated annealing operation is merged into the cuckoo search algorithm to enhance the local search ability and improve the accuracy and reliability of the results. The functionality of the proposed hybrid algorithm is investigated through the Lorenz chaotic system under the noiseless and noise condition, respectively. The numerical results demonstrate that the method can estimate parameters efficiently and accurately in the noiseless and noise condition. Finally, the results are compared with the traditional cuckoo search algorithm, genetic algorithm, and particle swarm optimization algorithm. Simulation results demonstrate the effectiveness and superior performance of the proposed algorithm.
Dynamics of quantum trajectories in chaotic systems
Wisniacki, D A; Benito, R M
2003-01-01
Quantum trajectories defined in the de Broglie--Bohm theory provide a causal way to interpret physical phenomena. In this Letter, we use this formalism to analyze the short time dynamics induced by unstable periodic orbits in a classically chaotic system, a situation in which scars are known to play a very important role. We find that the topologies of the quantum orbits are much more complicated than that of the scarring and associated periodic orbits, since the former have quantum interference built in. Thus scar wave functions are necessary to analyze the corresponding dynamics. Moreover, these topologies imply different return routes to the vicinity of the initial positions, and this reflects in the existence of different contributions in each peak of the survival probability function.
Directory of Open Access Journals (Sweden)
Alireza Khosravi
2012-03-01
Full Text Available This paper deals with the design of optimal backstepping controller, by using the chaotic particle swarm optimization (CPSO algorithm to control of chaos in Lure like chaotic system. The backstepping method consists of parameters which could have positive values. The parameters are usually chosen optional by trial and error method. The controlled system provides different behaviors for different values of the parameters. It is necessary to select proper parameters to obtain a good response, because the improper selection of the parameters leads to inappropriate responses or even may lead to instability of the system. The proposed optimal backstepping controller without trial and error determines the parameters of backstepping controller automatically and intelligently by minimizing the Integral of Time multiplied Absolute Error (ITAE and squared controller output. Finally, the efficiency of the proposed optimal backstepping controller (OBSC is illustrated by implementing the method on the Lure like chaotic system.
基于函数矩阵的一类混沌系统同步%Synchronizing chaotic systems based on an arbitrary function matrix
Institute of Scientific and Technical Information of China (English)
林立雄; 彭侠夫
2014-01-01
研究了一类混沌系统的同步问题、基于稳定性理论和极点配置技术，设计了两个混沌系统之间的同步方案，实现两个混沌系统之间的同步。通过函数矩阵，实现驱动系统和响应系统的状态变量按给定的函数矩阵同步。同时证明了该方法同样适用于两个混沌系统之间的滞后同步。通过对Lorenz混沌系统和Lorenz超混沌系统的数值模拟，进一步验证了所提方案的有效性。%In this paper, we introduce a type of chaotic synchronization, where two chaotic systems are synchronized based on a function matrix. In particular, each drive system state synchronizes with a certain combination of response system states. Based on linear system theory and the pole placement technique, the scheme is given and illustrated with hyperchaotic Lorenz system and Lorenz chaotic system. Numerical simulations are carried out to verify the effectiveness of the proposed scheme.
Passive control of Permanent Magnet Synchronous Motor chaotic systems
Institute of Scientific and Technical Information of China (English)
QI Dong-lian; WANG Jia-jun; ZHAO Guang-zhou
2005-01-01
Permanent Magnet Synchronous Motor model can exhibit a variety of chaotic phenomena under some choices of system parameters and external input. Based on the property of passive system, the essential conditions were studied, by which Permanent Magnet Synchronous Motor chaotic system could be equivalent to passive system. Using Lyapunov stability theory, the convergence condition deciding the system's characters was discussed. In the convergence condition area, the equivalent passive system could be globally asymptotically stabilized by smooth state feedback.
Synchronization of chaotic fractional-order systems via linear control
Odibat, Zaid,; Corson, Nathalie; Aziz-Alaoui, Moulay; Bertelle, Cyrille
2010-01-01
International audience; The chaotic dynamics of fractional-order systems has attracted much attention recently. Chaotic synchronization of fractional-order systems is further studied in this paper. We investigate the chaos synchronization of two identical systems via a suitable linear controller applied to the response system. Based on the stability results of linear fractional-order systems, sufficient conditions for chaos synchronization of these systems are given. Control laws are derived ...
Control of Unknown Chaotic Systems Based on Neural Predictive Control
Institute of Scientific and Technical Information of China (English)
LIDong-Mei; WANGZheng-Ou
2003-01-01
We introduce the predictive control into the control of chaotic system and propose a neural network control algorithm based on predictive control. The proposed control system stabilizes the chaotic motion in an unknown chaotic system onto the desired target trajectory. The proposed algorithm is simple and its convergence speed is much higher than existing similar algorithms. The control system can control hyperchaos. We analyze the stability of the control system and prove the convergence property of the neural controller. The theoretic derivation and simulations demonstrate the effectiveness of the algorithm.
Security analysis and modification of a chaotic encryption system
Institute of Scientific and Technical Information of China (English)
崔光亮; 冯正进; 胡国杰
2004-01-01
A type of digital chaotic encryption system was proposed in Ref. [1] which uses a class of 1-D piecewise linear (PWL) map to realize chaotic encryption and decryption system through the inverse system approach. In the general with the input terminal. In this paper we show that this cryptosystem can not frustrate chosen-cipher text attack. A type of chaotic encryption system based on self-synchronizing stream cipher is proposed. This system can avoid chosen-cipher text attack and has higher security.
Statistical Properties of Lorenz-like Flows, Recent Developments and Perspectives
Araujo, Vitor; Galatolo, Stefano; Pacifico, Maria José
We comment on the mathematical results about the statistical behavior of Lorenz equations and its attractor, and more generally on the class of singular hyperbolic systems. The mathematical theory of such kind of systems turned out to be surprisingly difficult. It is remarkable that a rigorous proof of the existence of the Lorenz attractor was presented only around the year 2000 with a computer-assisted proof together with an extension of the hyperbolic theory developed to encompass attractors robustly containing equilibria. We present some of the main results on the statistical behavior of such systems. We show that for attractors of three-dimensional flows, robust chaotic behavior is equivalent to the existence of certain hyperbolic structures, known as singular-hyperbolicity. These structures, in turn, are associated with the existence of physical measures: in low dimensions, robust chaotic behavior for flows ensures the existence of a physical measure. We then give more details on recent results on the dynamics of singular-hyperbolic (Lorenz-like) attractors: (1) there exists an invariant foliation whose leaves are forward contracted by the flow (and further properties which are useful to understand the statistical properties of the dynamics); (2) there exists a positive Lyapunov exponent at every orbit; (3) there is a unique physical measure whose support is the whole attractor and which is the equilibrium state with respect to the center-unstable Jacobian; (4) this measure is exact dimensional; (5) the induced measure on a suitable family of cross-sections has exponential decay of correlations for Lipschitz observables with respect to a suitable Poincaré return time map; (6) the hitting time associated to Lorenz-like attractors satisfy a logarithm law; (7) the geometric Lorenz flow satisfies the Almost Sure Invariance Principle (ASIP) and the Central Limit Theorem (CLT); (8) the rate of decay of large deviations for the volume measure on the ergodic basin of
Chaotic carrier pulse position modulation communication system and method
Abarbanel, Henry D. I.; Larson, Lawrence E.; Rulkov, Nikolai F.; Sushchik, Mikhail M.; Tsimring, Lev S.; Volkovskii, Alexander R.
2001-01-01
A chaotic carrier pulse position modulation communication system and method is disclosed. The system includes a transmitter and receiver having matched chaotic pulse regenerators. The chaotic pulse regenerator in the receiver produces a synchronized replica of a chaotic pulse train generated by the regenerator in the transmitter. The pulse train from the transmitter can therefore act as a carrier signal. Data is encoded by the transmitter through selectively altering the interpulse timing between pulses in the chaotic pulse train. The altered pulse train is transmitted as a pulse signal. The receiver can detect whether a particular interpulse interval in the pulse signal has been altered by reference to the synchronized replica it generates, and can therefore detect the data transmitted by the receiver. Preferably, the receiver predicts the earliest moment in time it can expect a next pulse after observation of at least two consecutive pulses. It then decodes the pulse signal beginning at a short time before expected arrival of a pulse.
The chaotic "sculpting" of the Solar System
Tsiganis, K.
2006-01-01
The orbits of the large celestial bodies in our Solar System are stable for very long times, as can be shown by numerical simulation. This gives the erroneous impression of perpetual stability of the system. It is only when we study the orbital distribution of the numerous minor bodies in the Solar System that we discover the rich variety of complex dynamical processes that have in fact shaped our system. During the last decade, enormous progress has been made, in understanding the evolution of the system over the last ~3.9 Gy. However, it also became clear that, in order to unveil its behaviour during the first ~700 million years of its lifetime, we have to find convincing explanations for observations that appear as details of its dynamical architecture. In the following we are going to show how the two best known - and up to now unexplained - observations in the Solar System, namely (i) the heavily cratered surface of the Moon and (ii) the elliptic (and not circular) motion of the planets, lead us to the discovery of the chaotic sculpting of the Solar System [1]-[3].
Chaotic and hyperchaotic attractors of a complex nonlinear system
Energy Technology Data Exchange (ETDEWEB)
Mahmoud, Gamal M; Al-Kashif, M A; Farghaly, A A [Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516 (Egypt)
2008-02-08
In this paper, we introduce a complex nonlinear hyperchaotic system which is a five-dimensional system of nonlinear autonomous differential equations. This system exhibits both chaotic and hyperchaotic behavior and its dynamics is very rich. Based on the Lyapunov exponents, the parameter values at which this system has chaotic, hyperchaotic attractors, periodic and quasi-periodic solutions and solutions that approach fixed points are calculated. The stability analysis of these fixed points is carried out. The fractional Lyapunov dimension of both chaotic and hyperchaotic attractors is calculated. Some figures are presented to show our results. Hyperchaos synchronization is studied analytically as well as numerically, and excellent agreement is found.
Equilibrium points and bifurcation control of a chaotic system
Institute of Scientific and Technical Information of China (English)
Liang Cui-Xiang; Tang Jia-Shi
2008-01-01
Based on the Routh-Hurwitz criterion,this paper investigates the stability of a new chaotic system.State feedback controllers are designed to control the chaotic system to the unsteady equilibrium points and limit cycle.Theoretical analyses give the range of value of control parameters to stabilize the unsteady equilibrium points of the chaotic system and its critical parameter for generating Hopf bifurcation.Certain nP periodic orbits can be stabilized by parameter adjustment.Numerical simulations indicate that the method can effectively guide the system trajectories to unsteady equilibrium points and periodic orbits.
Complexity analyses of multi-wing chaotic systems
Institute of Scientific and Technical Information of China (English)
He Shao-Bo; Sun Ke-Hui; Zhu Cong-Xu
2013-01-01
The complexities of multi-wing chaotic systems based on the modified Chen system and a multi-segment quadratic function are investigated by employing the statistical complexity measure (SCM) and the spectral entropy (SE) algorithm.How to choose the parameters of the SCM and SE algorithms is discussed.The results show that the complexity of the multi-wing chaotic system does not increase as the number of wings increases,and it is consistent with the results of the Grassberger-Procaccia (GP) algorithm and the largest Lyapunov exponent (LLE) of the multi-wing chaotic system.
Complexity analyses of multi-wing chaotic systems
He, Shao-Bo; Sun, Ke-Hui; Zhu, Cong-Xu
2013-05-01
The complexities of multi-wing chaotic systems based on the modified Chen system and a multi-segment quadratic function are investigated by employing the statistical complexity measure (SCM) and the spectral entropy (SE) algorithm. How to choose the parameters of the SCM and SE algorithms is discussed. The results show that the complexity of the multi-wing chaotic system does not increase as the number of wings increases, and it is consistent with the results of the Grassberger—Procaccia (GP) algorithm and the largest Lyapunov exponent (LLE) of the multi-wing chaotic system.
Generalized Synchronization Between Different Fractional-Order Chaotic Systems
Institute of Scientific and Technical Information of China (English)
ZHOU Ping; CHENG Xue-Feng; ZHANG Nian-Ying
2008-01-01
In this paper, using scalar feedback controller and stability theory of fractional-order systems, a gener-alized synchronization method for different fractional-order chaotic systems is established. Simulation results show the effectiveness of the theoretical results.
Chaotic synchronization for a class of fractional-order chaotic systems
Institute of Scientific and Technical Information of China (English)
Zhou Ping
2007-01-01
In this paper, a very simple synchronization method is presented for a class of fractional-order chaotic systems only via feedback control. The synchronization technique, based on the stability theory of fractional-order systems, is simple and theoretically rigorous.
An optical CDMA system based on chaotic sequences
Liu, Xiao-lei; En, De; Wang, Li-guo
2014-03-01
In this paper, a coherent asynchronous optical code division multiple access (OCDMA) system is proposed, whose encoder/decoder is an all-optical generator. This all-optical generator can generate analog and bipolar chaotic sequences satisfying the logistic maps. The formula of bit error rate (BER) is derived, and the relationship of BER and the number of simultaneous transmissions is analyzed. Due to the good property of correlation, this coherent OCDMA system based on these bipolar chaotic sequences can support a large number of simultaneous users, which shows that these chaotic sequences are suitable for asynchronous OCDMA system.
Spread spectrum communication system with chaotic frequency modulation
Volkovskii, A. R.; Tsimring, L. Sh.; Rulkov, N. F.; Langmore, I.
2005-09-01
A new spread spectrum communication system utilizing chaotic frequency modulation of sinusoidal signals is discussed. A single phase lock loop (PLL) system in the receiver is used both to synchronize the local chaotic oscillator and to recover the information signal. We study the dynamics of the synchronization process, stability of the PLL system, and evaluate the bit-error-rate performance of this chaos-based communication system.
Passive control of a 4-scroll chaotic system
Institute of Scientific and Technical Information of China (English)
Wang Fa-Qiang; Liu Chong-Xin
2007-01-01
This paper studies the control of a new chaotic system which can generate 4-scroll attractors. Based on the properties of a passive system, it derives the essential conditions under which this new chaotic system could be equivalent to a passive system and globally asymptotically stabilize at a zero equilibrium point via smooth state feedback. Simulation results and circuit experiment show that the proposed chaos control method is effective.
CHAOTIC DISINTEGRATION OF THE INNER SOLAR SYSTEM
Energy Technology Data Exchange (ETDEWEB)
Batygin, Konstantin [Division of Geological and Planetary Sciences, California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125 (United States); Morbidelli, Alessandro [Department Lagrange, Observatoire de la Côte d' Azur, F-06304 Nice (France); Holman, Mathew J. [Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 (United States)
2015-02-01
On timescales that greatly exceed an orbital period, typical planetary orbits evolve in a stochastic yet stable fashion. On even longer timescales, however, planetary orbits can spontaneously transition from bounded to unbound chaotic states. Large-scale instabilities associated with such behavior appear to play a dominant role in shaping the architectures of planetary systems, including our own. Here we show how such transitions are possible, focusing on the specific case of the long-term evolution of Mercury. We develop a simple analytical model for Mercury's dynamics and elucidate the origins of its short-term stochastic behavior as well as of its sudden progression to unbounded chaos. Our model allows us to estimate the timescale on which this transition is likely to be triggered, i.e., the dynamical lifetime of the solar system as we know it. The formulated theory is consistent with the results of numerical simulations and is broadly applicable to extrasolar planetary systems dominated by secular interactions. These results constitute a significant advancement in our understanding of the processes responsible for sculpting of the dynamical structures of generic planetary systems.
Impact of hyperbolicity on chimera states in ensembles of nonlocally coupled chaotic oscillators
Energy Technology Data Exchange (ETDEWEB)
Semenova, N.; Anishchenko, V. [Department of Physics, Saratov State University, Astrakhanskaya Str. 83, 410012 Saratov (Russian Federation); Zakharova, A.; Schöll, E. [Institut für Theoretische Physik, TU Berlin, Hardenbergstraße 36, 10623 Berlin (Germany)
2016-06-08
In this work we analyse nonlocally coupled networks of identical chaotic oscillators. We study both time-discrete and time-continuous systems (Henon map, Lozi map, Lorenz system). We hypothesize that chimera states, in which spatial domains of coherent (synchronous) and incoherent (desynchronized) dynamics coexist, can be obtained only in networks of chaotic non-hyperbolic systems and cannot be found in networks of hyperbolic systems. This hypothesis is supported by numerical simulations for hyperbolic and non-hyperbolic cases.
Modified scaling function projective synchronization of chaotic systems
Institute of Scientific and Technical Information of China (English)
Xu Yu-Hua; Zhou Wu-Neng; Fang Jian-An
2011-01-01
This paper investigates a kind of modified scaling function projective synchronization of uncertain chaotic systems using an adaptive controller. The given scaling function in the new method can be an equilibrium point,a periodic orbit,or even a chaotic attractor in the phase space. Based on LaSa11e's invariance set principle,the adaptive control law is derived to make the states of two chaotic systems function projective synchronized. Some numerical examples are also given to show the effectiveness of the proposed method.
Exploiting the natural redundancy of chaotic signals in communication systems
Marino; Rosa; Grebogi
2000-09-18
Chaotic signals can be used as carriers of information in communication systems. In this work we describe a simple encoding method that allows one to map any desired bit sequence into a chaotic waveform. The redundancy of the resulting information carrying signal enables us to devise a novel signal reconstruction technique that is able to recover relatively large parts of the chaotic signal starting from just a few samples of it. We show that this technique allows one to increase both the transmission reliability and the transmission rate of a communication system even in the presence of noise.
Linear generalized synchronization of continuous-time chaotic systems
Energy Technology Data Exchange (ETDEWEB)
Lu Junguo E-mail: jglu@sjtu.edu.cn; Xi Yugeng
2003-08-01
This paper develops a general approach for constructing a response system to implement linear generalized synchronization (GS) with the drive continuous-time chaotic system. Some sufficient conditions of global asymptotic linear GS between the drive and response continuous-time chaotic systems are attained from rigorously modern control theory. Finally, we take Chua's circuit as an example for illustration and verification.
Passive control of chaotic system with multiple strange attractors
Institute of Scientific and Technical Information of China (English)
Song Yun-Zhong; Zhao Guang-Zhou; Qi Dong-Lian
2006-01-01
In this paper we present a new simple controller for a chaotic system, that is, the Newton-Leipnik equation with two strange attractors: the upper attractor (UA) and the lower attractor (LA). The controller design is based on the passive technique. The final structure of this controller for original stabilization has a simple nonlinear feedback form.Using a passive method, we prove the stability of a closed-loop system. Based on the controller derived from the passive principle, we investigate three different kinds of chaotic control of the system, separately: the original control forcing the chaotic motion to settle down to the origin from an arbitrary position of the phase space; the chaotic intra-attractor control for stabilizing the equilibrium points only belonging to the upper chaotic attractor or the lower chaotic one,and the inter-attractor control for compelling the chaotic oscillation from one basin to another one. Both theoretical analysis and simulation results verify the validity of the suggested method.
An Adaptive Tracking Control of Fractional-Order Chaotic Systems with Uncertain System Parameter
Ping Zhou; Rui Ding
2011-01-01
An adaptive tracking control scheme is presented for fractional-order chaotic systems with uncertain parameter. It is theoretically proved that this approach can make the uncertain parameter fractional-order chaotic system track any given reference signal and the uncertain system parameter is estimated through the adaptive tracking control process. Furthermore, the reference signal may belong to other integer-orders chaotic system or belong to different fractional-order chaotic system with di...
Decoherence, delocalization and irreversibility in quantum chaotic systems
Shiokawa, K; Shiokawa, K; Hu, B L
1995-01-01
Decoherence in quantum systems which are classically chaotic is studied. The Arnold cat map and the quantum kicked rotor are chosen as examples of linear and nonlinear chaotic systems. The Feynman-Vernon influence functional formalism is used to study the effect of the environment on the system. It is well-known that quantum coherence can obliterate many chaotic behavior in the corresponding classical system. But interaction with an environment can under general circumstances quickly diminish quantum coherence and reenact many classical chaotic behavior. How effective decoherence works to sustain chaos, and how the resultant behavior qualitatively differs from the quantum picture depend on the coupling of the system with the environment and the spectral density and temperature of the environment. We show how recurrence in the quantum cat map is lost and classical ergodicity is recovered due to the effect of the environment. Quantum coherence and diffusion suppression are instrumental to dynamical localization...
Chaotic synchronization in coupled spatially extended beam-plasma systems
Filatov, Roman A.; Hramov, Alexander E.; ALEXEY A. KORONOVSKII
2006-01-01
The appearance of the chaotic synchronization regimes has been discovered for the coupled spatially extended beam-plasma Pierce systems. The coupling was introduced only on the right bound of each subsystem. It has been shown that with coupling increase the spatially extended beam-plasma systems show the transition from asynchronous behavior to the phase synchronization and then to the complete synchronization regime. For the consideration of the chaotic synchronization we used the concept of...
Impulsive control of chaotic systems with exogenous perturbations
Institute of Scientific and Technical Information of China (English)
Liu Xing-Wen; Huang Qin-Zhen; Gao Xin; Shao Shi-Quan
2007-01-01
The impulsive control of chaotic systems, which are subjected to unbounded exogenous perturbations, is considered. By using the theory of impulsive differential equation together with the fuzzy control technique, the authors propose an impulsive robust chaos controlling criterion expressed as linear matrix inequalities (LMIs). Based on the proposed control criterion, the procedure for designing impulsive controllers of common (perturbed) chaotic systems is provided. Finally, a numerical example is given to demonstrate the obtained theoretical result.
Blind adaptive identification of FIR channel in chaotic communication systems
Institute of Scientific and Technical Information of China (English)
Wang Bao-Yun; Tommy W.S.Chow; K.T.Ng
2004-01-01
In this paper we study the problem of blind channel identification in chaotic communications. An adaptive algorithm is proposed, which exploits the boundness property of chaotic signals. Compared with the EKF-based approach, the proposed algorithm achieves a great complexity gain but at the expense of a slight accuracy degradation.However, our approach enjoys the important advantage that it does not require the a priori information such as nonlinearity of chaotic dynamics and the variances of measurement noise and the coefficient model noise. In addition,our approach is applicable to the ARMA system.
Pattern formations in chaotic spatio-temporal systems
Indian Academy of Sciences (India)
Ying Zhang; Shihong Wang; Jinhua Xiao; Hilda A Cerdeira; S Chen; Gang Hu
2005-06-01
Pattern formations in chaotic spatio-temporal systems modelled by coupled chaotic oscillators are investigated. We focus on various symmetry breakings and different kinds of chaos synchronization–desynchronization transitions, which lead to certain types of spontaneous spatial orderings and the emergence of some typical ordered patterns, such as rotating wave patterns with splay phase ordering (orientational symmetry breaking) and partially synchronous standing wave patterns with in-phase ordering (translational symmetry breaking). General pictures of the global behaviors of pattern formations and transitions in coupled chaotic oscillators are provided.
Wigner function statistics in classically chaotic systems
Horvat, M; Horvat, Martin; Prosen, Tomaz
2003-01-01
We have studied statistical properties of the values of the Wigner function W(x) of 1D quantum maps on compact 2D phase space of finite area V. For this purpose we have defined a Wigner function probability distribution P(w) = (1/V) int delta(w-W(x)) dx, which has, by definition, fixed first and second moment. In particular, we concentrate on relaxation of time evolving quantum state in terms of W(x), starting from a coherent state. We have shown that for a classically chaotic quantum counterpart the distribution P(w) in the semi-classical limit becomes a Gaussian distribution that is fully determined by the first two moments. Numerical simulations have been performed for the quantum sawtooth map and the quantized kicked top. In a quantum system with Hilbert space dimension N (similar 1/hbar) the transition of P(w) to a Gaussian distribution was observed at times t proportional to log N. In addition, it has been shown that the statistics of Wigner functions of propagator eigenstates is Gaussian as well in the...
Generation and dynamics analysis of N-scrolls existence in new translation-type chaotic systems
Liu, Yue; Guo, Shuxu
2016-11-01
In this paper, we propose two kinds of translation type chaotic systems for creating 2 N + 1-and 2(N + 1)-scrolls chaotic attractors from a simple three-dimensional system, which are named the translation-2 chaotic system (a12a21 0). We also propose the successful design criterion for constructing 2 N + 1-and 2(N + 1)-scrolls, respectively. Then, the dynamics property of the translation-2 chaotic system is studied in detail. MATLAB simulation results show that very sophisticated dynamical behaviors and unique chaotic behaviors of the system. Finally, the definition and criterion of multi-scroll attractors for the translation-3 chaotic system is obtained. Three representative examples are shown in some classical chaotic systems that can be equally obtained via the set parameters of the translation type chaotic system. Furthermore, we show that the translation type chaotic systems have similar but topologically non-equivalent chaotic attractors, and they are the three-dimensional ordinary differential equations.
Efficiency of Monte Carlo sampling in chaotic systems.
Leitão, Jorge C; Lopes, J M Viana Parente; Altmann, Eduardo G
2014-11-01
In this paper we investigate how the complexity of chaotic phase spaces affect the efficiency of importance sampling Monte Carlo simulations. We focus on flat-histogram simulations of the distribution of finite-time Lyapunov exponent in a simple chaotic system and obtain analytically that the computational effort: (i) scales polynomially with the finite time, a tremendous improvement over the exponential scaling obtained in uniform sampling simulations; and (ii) the polynomial scaling is suboptimal, a phenomenon known as critical slowing down. We show that critical slowing down appears because of the limited possibilities to issue a local proposal in the Monte Carlo procedure when it is applied to chaotic systems. These results show how generic properties of chaotic systems limit the efficiency of Monte Carlo simulations.
Bifurcation diagrams in relation to synchronization in chaotic systems
Indian Academy of Sciences (India)
Debabrata Dutta; Sagar Chakraborty
2010-06-01
We numerically study some of the three-dimensional dynamical systems which exhibit complete synchronization as well as generalized synchronization to show that these systems can be conveniently partitioned into equivalent classes facilitating the study of bifurcation diagrams within each class. We demonstrate how bifurcation diagrams may be helpful in predicting the nature of the driven system by knowing the bifurcation diagram of driving system and vice versa. The study is extended to include the possible generalized synchronization between elements of two different equivalent classes by taking the Rössler-driven-Lorenz-system as an example.
Chaotic diffusion in the Gliese-876 planetary system
Martí, J. G.; Cincotta, P. M.; Beaugé, C.
2016-07-01
Chaotic diffusion is supposed to be responsible for orbital instabilities in planetary systems after the dissipation of the protoplanetary disc, and a natural consequence of irregular motion. In this paper, we show that resonant multiplanetary systems, despite being highly chaotic, not necessarily exhibit significant diffusion in phase space, and may still survive virtually unchanged over time-scales comparable to their age. Using the GJ-876 system as an example, we analyse the chaotic diffusion of the outermost (and less massive) planet. We construct a set of stability maps in the surrounding regions of the Laplace resonance. We numerically integrate ensembles of close initial conditions, compute Poincaré maps and estimate the chaotic diffusion present in this system. Our results show that, the Laplace resonance contains two different regions: an inner domain characterized by low chaoticity and slow diffusion, and an outer one displaying larger values of dynamical indicators. In the outer resonant domain, the stochastic borders of the Laplace resonance seem to prevent the complete destruction of the system. We characterize the diffusion for small ensembles along the parameters of the outermost planet. Finally, we perform a stability analysis of the inherent chaotic, albeit stable Laplace resonance, by linking the behaviour of the resonant variables of the configurations to the different sub-structures inside the three-body resonance.
Reconstruction of time-delay systems from chaotic time series.
Bezruchko, B P; Karavaev, A S; Ponomarenko, V I; Prokhorov, M D
2001-11-01
We propose a method that allows one to estimate the parameters of model scalar time-delay differential equations from time series. The method is based on a statistical analysis of time intervals between extrema in the time series. We verify our method by using it for the reconstruction of time-delay differential equations from their chaotic solutions and for modeling experimental systems with delay-induced dynamics from their chaotic time series.
Reservoir observers: Model-free inference of unmeasured variables in chaotic systems.
Lu, Zhixin; Pathak, Jaideep; Hunt, Brian; Girvan, Michelle; Brockett, Roger; Ott, Edward
2017-04-01
Deducing the state of a dynamical system as a function of time from a limited number of concurrent system state measurements is an important problem of great practical utility. A scheme that accomplishes this is called an "observer." We consider the case in which a model of the system is unavailable or insufficiently accurate, but "training" time series data of the desired state variables are available for a short period of time, and a limited number of other system variables are continually measured. We propose a solution to this problem using networks of neuron-like units known as "reservoir computers." The measurements that are continually available are input to the network, which is trained with the limited-time data to output estimates of the desired state variables. We demonstrate our method, which we call a "reservoir observer," using the Rössler system, the Lorenz system, and the spatiotemporally chaotic Kuramoto-Sivashinsky equation. Subject to the condition of observability (i.e., whether it is in principle possible, by any means, to infer the desired unmeasured variables from the measured variables), we show that the reservoir observer can be a very effective and versatile tool for robustly reconstructing unmeasured dynamical system variables.
Institute of Scientific and Technical Information of China (English)
张解放; 郑春龙; 孟剑平; 方建平
2003-01-01
With the help of variable separation approach, a quite general excitation of a new (2+l)-dimensional long dispersive wave system is derived. The chaotic behaviour, such as chaotic line soliton patterns, chaotic dromion patterns, chaotic-period patterns, and chaotic-chaotic patterns, in some new localized excitations are found by selecting appropriate functions.
Chaos Synchronization on Parameters Adaptive Control for Chen Chaotic System
Institute of Scientific and Technical Information of China (English)
ZHOU Ping
2003-01-01
Chaos synchronization of Chen chaotic system for parameters unknown is discussed in this paper using a scalar output. Using the concept of conditional Lyapunov exponents, the negativity of all Lyapunov exponents shows the synchronization of transmitter systems with receiver systems even though system parametes are not known to receiver systems.
Modelling and Research of Chaotic Rossler System with LabView and Multisim Software Environment
Directory of Open Access Journals (Sweden)
V. B. Rusyn
2014-12-01
Full Text Available Introduction. In this paper is presented a theoretical basis of chaotic Rossler system. Modelling of Chaotic Rossler System in LabView. Submitted programming interface that has been developed in LabView software environment. It allows generating and researching chaotic Rossler system. Submitted by time distribution of three chaotic coordinates and spectral analysis. Also submitted values of variables in which generated different period (controlled attractors of the chaotic Rossler system. The software interface demonstrates masking and decrypt information carrier of the chaotic Rossler system. Modelling of Chaotic Rossler System in MultiSim. Using MultiSim software environment conducted scheme technical analysis circuit of a generator that implements a chaotic Rossler system. Conclusions. Modelled circuit of generator confirming correspondence scheme-technical solution to mathematical apparatus that describing chaotic Rossler system. Keywords: chaos; control; system; Rossler; LabView; MultiSim
Control of Unknown Chaotic Systems Based on Neural Predictive Control
Institute of Scientific and Technical Information of China (English)
LI Dong-Mei; WANG Zheng-Ou
2003-01-01
We introduce the predictive control into the control of chaotic system and propose a neural networkcontrol algorithm based on predictive control. The proposed control system stabilizes the chaotic motion in an unknownchaotic system onto the desired target trajectory. The proposed algorithm is simple and its convergence speed is muchhigher than existing similar algorithms. The control system can control hyperchaos. We analyze the stability of thecontrol system and prove the convergence property of the neural controller. The theoretic derivation and simulationsdemonstrate the effectiveness of the algorithm.
Realization of fractional-order Liu chaotic system by circuit
Institute of Scientific and Technical Information of China (English)
Lu Jun-Jie; Liu Chong-Xin
2007-01-01
In this paper, chaotic behaviours in the fractional-order Liu system are studied. Based on the approximation theory of fractional-order operator, circuits are designed to simulate the fractional- order Liu system with q = 0.1 - 0.9 in a step of 0.1, and an experiment has demonstrated the 2.7-order Liu system. The simulation results prove that the chaos exists indeed in the fractional-order Liu system with an order as low as 0.3. The experimental results prove that the fractional-order chaotic system can be realized by using hardware devices, which lays the foundation for its practical applications.
Inhomogeneous stationary and oscillatory regimes in coupled chaotic oscillators.
Liu, Weiqing; Volkov, Evgeny; Xiao, Jinghua; Zou, Wei; Zhan, Meng; Yang, Junzhong
2012-09-01
The dynamics of linearly coupled identical Lorenz and Pikovsky-Rabinovich oscillators are explored numerically and theoretically. We concentrate on the study of inhomogeneous stable steady states ("oscillation death (OD)" phenomenon) and accompanying periodic and chaotic regimes that emerge at an appropriate choice of the coupling matrix. The parameters, for which OD occurs, are determined by stability analysis of the chosen steady state. Three model-specific types of transitions to and from OD are observed: (1) a sharp transition to OD from a nonsymmetric chaotic attractor containing random intervals of synchronous chaos; (2) transition to OD from the symmetry-breaking chaotic regime created by negative coupling; (3) supercritical bifurcation of OD into inhomogeneous limit cycles and further evolution of the system to inhomogeneous chaotic regimes that coexist with complete synchronous chaos. These results may fill a gap in the understanding of the mechanism of OD in coupled chaotic systems.
Noise Separation from the Weak Signal Chaotic Detection System
Directory of Open Access Journals (Sweden)
Hanjie Gu
2014-11-01
Full Text Available The traditional weak signal chaotic detection system still restricts some technical issues in the situation of the signal with noise, such as poor denoising ability and low detection precision. In this paper, we propose a novel weak signal chaotic detection system based on an improved wavelet transform algorithm. First, the traditional wavelet transform algorithm domain variables have been transformed and discretized to eliminate the redundant transform. Then, based on the discrete optimization, the wavelet coefficients have been optimized by threshold compromise strategy. The improved wavelet transform algorithm is applied in the weak signal chaotic detection system. The noise signal after finite discrete processing is treated as a perturbation of cycle power and put into a chaotic system for detecting weak signal under the noise conditions. The simulation experiments show that the proposed improved wavelet transform algorithm has a better denoising effect than the traditional wavelet transform algorithm. Moreover, the improved algorithm shows better accuracy and higher robustness in the weak signal chaotic detection system.
Optimal Control for a Class of Chaotic Systems
Directory of Open Access Journals (Sweden)
Jianxiong Zhang
2012-01-01
Full Text Available This paper proposes the optimal control methods for a class of chaotic systems via state feedback. By converting the chaotic systems to the form of uncertain piecewise linear systems, we can obtain the optimal controller minimizing the upper bound on cost function by virtue of the robust optimal control method of piecewise linear systems, which is cast as an optimization problem under constraints of bilinear matrix inequalities (BMIs. In addition, the lower bound on cost function can be achieved by solving a semidefinite programming (SDP. Finally, numerical examples are given to illustrate the results.
Adaptive synchronization of Chen chaotic system with uncertain parameters
Institute of Scientific and Technical Information of China (English)
LIU Yu-liang; ZHU Jie; DING Da-wei
2007-01-01
By the control method independent of system parameters, synchronization of two Chen chaotic systems with identical, but uncertain parameters is discussed in this article. Based on Lyapunov's theorem, an adaptive controller and the parameters estimate update law are derived to make the solution of the error dynamical equation converge at the point E(0, 0, 0) with a quick speed, that is, the states of two Chen chaotic systems can be asymptotically synchronized. Numerical simulations verify the effectiveness and the adaptivity to the variation of Chen system parameters.
Mixed coherent states in coupled chaotic systems: Design of secure wireless communication
Vigneshwaran, M.; Dana, S. K.; Padmanaban, E.
2016-12-01
A general coupling design is proposed to realize a mixed coherent (MC) state: coexistence of complete synchronization, antisynchronization, and amplitude death in different pairs of similar state variables of the coupled chaotic system. The stability of coupled system is ensured by the Lyapunov function and a scaling of each variable is also separately taken care of. When heterogeneity as a parameter mismatch is introduced in the coupled system, the coupling function facilitates to retain its coherence and displays the global stability with renewed scaling factor. Robust synchronization features facilitated by a MC state enable to design a dual modulation scheme: binary phase shift key (BPSK) and parameter mismatch shift key (PMSK), for secure data transmission. Two classes of decoders (coherent and noncoherent) are discussed, the noncoherent decoder shows better performance over the coherent decoder, mostly a noncoherent demodulator is preferred in biological implant applications. Both the modulation schemes are demonstrated numerically by using the Lorenz oscillator and the BPSK scheme is demonstrated experimentally using radio signals.
Compact Global Chaotic Attractors of Discrete Control Systems
Directory of Open Access Journals (Sweden)
Cheban David
2014-01-01
Full Text Available The paper is dedicated to the study of the problem of existence of compact global chaotic attractors of discrete control systems and to the description of its structure. We consider so called switched systems with discrete time xn+1 = fv(n(xn, where v: Z+ → {1; 2; : : : ;m}. If m≥2 we give sufficient conditions (the family M := {f1; f2; : : : ; fm} of functions is contracting in the extended sense for the existence of a compact global chaotic attractor. We study this problem in the framework of non-autonomous dynamical systems (cocycles
Synchronization of Unified Chaotic System Using Occasional Driving
Institute of Scientific and Technical Information of China (English)
Wu Xiao-qun; Lu Jun-an
2003-01-01
The synchronization of the unified chaotic systems using occasional driving technique is studied. The relation among interim period T, sampling interval 2ε, feedback gain r and the parameter α of the system is thoroughly investigated. Numerical results show that smaller interim period T and properly larger sampling interval 2ε can accelerate the synchronizing pace. Furthermore, for a unified chaotic system in which a is given, we can achieve satisfying synchronizing results as long as T,ε and r are appropriately chosen. As we adopt the occasional driving method, we greatly reduce the control cost. Therefore with this method we can obtain the expecting goals with little control cost.
Automatic control and tracking of periodic orbits in chaotic systems.
Ando, Hiroyasu; Boccaletti, S; Aihara, Kazuyuki
2007-06-01
Based on an automatic feedback adjustment of an additional parameter of a dynamical system, we propose a strategy for controlling periodic orbits of desired periods in chaotic dynamics and tracking them toward the set of unstable periodic orbits embedded within the original chaotic attractor. The method does not require information on the system to be controlled, nor on any reference states for the targets, and it overcomes some of the difficulties encountered by other techniques. Assessments of the method's effectiveness and robustness are given by means of the application of the technique to the stabilization of unstable periodic orbits in both discrete- and continuous-time systems.
Active synchronization between two different chaotic dynamical system
Energy Technology Data Exchange (ETDEWEB)
Maheri, M. [Institute for Mathematical Research, 43400 UPM, Serdang, Selengor (Malaysia); Arifin, N. Md; Ismail, F. [Department of Mathematics, 43400 UPM, Serdang, Selengor (Malaysia)
2015-05-15
In this paper we investigate on the synchronization problem between two different chaotic dynamical system based on the Lyapunov stability theorem by using nonlinear control functions. Active control schemes are used for synchronization Liu system as drive and Rossler system as response. Numerical simulation by using Maple software are used to show effectiveness of the proposed schemes.
The control of an optical hyper-chaotic system
Energy Technology Data Exchange (ETDEWEB)
Jiang Shumin [Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013 (China); Tian Lixin [Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013 (China)]. E-mail: tianlx@ujs.edu.cn; Wang Xuedi [Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013 (China)
2007-12-15
This paper discusses the problem of hyper-chaos control of an optical system. Based on Lyapunov stability theory, a non-autonomous feedback controller is designed. The proposed controller ensures that the hyper-chaotic system will be asymptotically stable. Numerical simulation of the original and the controlled system is provided to show the effectiveness of our method.
Directory of Open Access Journals (Sweden)
Lili Lei
2012-05-01
Full Text Available A hybrid data assimilation approach combining nudging and the ensemble Kalman filter (EnKF for dynamic analysis and numerical weather prediction is explored here using the non-linear Lorenz three-variable model system with the goal of a smooth, continuous and accurate data assimilation. The hybrid nudging-EnKF (HNEnKF computes the hybrid nudging coefficients from the flow-dependent, time-varying error covariance matrix from the EnKF's ensemble forecasts. It extends the standard diagonal nudging terms to additional off-diagonal statistical correlation terms for greater inter-variable influence of the innovations in the model's predictive equations to assist in the data assimilation process. The HNEnKF promotes a better fit of an analysis to data compared to that achieved by either nudging or incremental analysis update (IAU. When model error is introduced, it produces similar or better root mean square errors compared to the EnKF while minimising the error spikes/discontinuities created by the intermittent EnKF. It provides a continuous data assimilation with better inter-variable consistency and improved temporal smoothness than that of the EnKF. Data assimilation experiments are also compared to the ensemble Kalman smoother (EnKS. The HNEnKF has similar or better temporal smoothness than that of the EnKS, and with much smaller central processing unit (CPU time and data storage requirements.
Design of output feedback controller for a unified chaotic system
Institute of Scientific and Technical Information of China (English)
Li Wen-Lin; Chen Xiu-Qin; Shen Zhi-Ping
2008-01-01
In this paper,the synchronization of a unified chaotic system is investigated by the use of output feedback controllers;a two-input single-output feedback controller and single-input single-output feedback controller are presented to synchronize the unified chaotic system when the states are not all measurable.Compared with the existing results,the controllers designed in this paper have some advantages such as small feedback gain,simple structure and less conservation.Finally,numerical simulations results are provided to demonstrate the validity and effectiveness of the proposed method.
Synchronization of an uncertain chaotic system via recurrent neural networks
Institute of Scientific and Technical Information of China (English)
谭文; 王耀南
2005-01-01
Incorporating distributed recurrent networks with high-order connections between neurons, the identification and synchronization problem of an unknown chaotic system in the presence of unmodelled dynamics is investigated. Based on the Lyapunov stability theory, the weights learning algorithm for the recurrent high-order neural network model is presented. Also, analytical results concerning the stability properties of the scheme are obtained. Then adaptive control law for eliminating synchronization error of uncertain chaotic plant is developed via Lyapunov methodology.The proposed scheme is applied to model and synchronize an unknown Rossler system.
Synchronization of Unified Chaotic System Using Occasional Driving
Institute of Scientific and Technical Information of China (English)
WuXiao-qun; LuJun-an
2003-01-01
The synchronization of the unified chaotic systems using occasional driving technique is studied. The relation among interim period T, sampling interval 2ε, feedback gain r and the parameter α of the system is thoroughly investigated. Numerical results show that smaller interim period T and properly larger sampling interval 2ε can accelerate the synchronizing pace. Furthermore, for a unified chaotic systemin which α is given, we can achieve satisfying synchronizing results as long as T,ε and r are appropriately chosen. As we adopt the occasional driving method, we greatly reduce the control cost. Therefore with this method we can obtain the expeetlng goals with little control cost.
Fuzzy Control of Chaotic System with Genetic Algorithm
Institute of Scientific and Technical Information of China (English)
FANG Jian-an; GUO Zhao-xia; SHAO Shi-huang
2002-01-01
A novel approach to control the unpredictable behavior of chaotic systems is presented. The control algorithm is based on fuzzy logic control technique combined with genetic algorithm. The use of fuzzy logic allows for the implementation of human "rule-of-thumb" approach to decision making by employing linguistic variables. An improved Genetic Algorithm (GA) is used to learn to optimally select the fuzzy membership functions of the linguistic labels in the condition portion of each rule,and to automatically generate fuzzy control actions under each condition. Simulation results show that such an approach for the control of chaotic systems is both effective and robust.
Stable classical structures in dissipative quantum chaotic systems
Raviola, Lisandro A; Rivas, Alejandro M F
2009-01-01
We study the stability of classical structures in chaotic systems when a dissipative quantum evolution takes place. We consider a paradigmatic model, the quantum baker map in contact with a heat bath at finite temperature. We analyze the behavior of the purity, fidelity and Husimi distributions corresponding to initial states localized on short periodic orbits (scar functions) and map eigenstates. Scar functions, that have a fundamental role in the semiclassical description of chaotic systems, emerge as very robust against environmental perturbations. This is confirmed by the study of other states localized on classical structures. Also, purity and fidelity show a complementary behavior as decoherence measures.
Hybrid synchronization of two independent chaotic systems on complex network
Indian Academy of Sciences (India)
NIAN FUZHONG; LIU WEILONG
2016-06-01
The real network nodes are always interfered by other messages. So, how to realize the hybrid synchronization of two independent chaotic systems based on the complex network is very important. To solve this problem, two other problems should be considered. One is how the same network node of the complex network was affected by different information sources. Another is how to achieve hybrid synchronization on the network. In this paper, the theoretical analysis andnumerical simulation on various complex networks are implemented. The results indicate that the hybrid synchronization of two independent chaotic systems is feasible.
A novel four-wing non-equilibrium chaotic system and its circuit implementation
Indian Academy of Sciences (India)
Lin Yuan; Wang Chunhua; He Haizhen; Zhou Li Li
2016-04-01
In this paper, we construct a novel, 4D smooth autonomous system. Compared to the existing chaotic systems, the most attractive point is that this system does not display any equilibria, but can still exhibit four-wing chaotic attractors. The proposed system is investigated through numerical simulations and analyses including time phase portraits, Lyapunov exponents, bifurcation diagram, and Poincaré maps. There is little difference between this chaotic system withoutequilibria and other chaotic systems with equilibria shown by phase portraits and Lyapunov exponents. But the bifurcation diagram shows that the chaotic systems without equilibria do not have characteristics such as pitchfork bifurcation, Hopf bifurcation etc. which are common to the normal chaotic systems. The Poincaré maps show that this system is a four-wing chaotic system with more complicated dynamics. Moreover, the physical existence of the four-wing chaotic attractor without equilibria is verified by an electronic circuit.
Synchronization between two different chaotic systems with nonlinear feedback control
Institute of Scientific and Technical Information of China (English)
Lü Ling; Guo Zhi-An; Zhang Chao
2007-01-01
This paper presents chaos synchronization between two different chaotic systems by using a nonlinear controller, in which the nonlinear functions of the system are used as a nonlinear feedback term. The feedback controller is designed on the basis of stability theory, and the area of feedback gain is determined. The artificial simulation results show that this control method is commendably effective and feasible.
Chaotic behaviour of a predator-prey system
Kooi, B.W.; Boer, M.P.
2003-01-01
Generally a predator-prey system is modelled by two ordinary differential equations which describe the rate of changes of the biomasses. Since such a system is two-dimensional no chaotic behaviour can occur. In the popular Rosenzweig-MacArthur model, which replaced the Lotka-Volterra model, a stable
A universal projective synchronization of general autonomous chaotic system
Indian Academy of Sciences (India)
Fuzhong Nian; Zingyuan Wang; Ming Li; Ge Guo
2012-12-01
This paper investigates the generalized projective synchronization in general autonomous chaotic system. A universal controller is designed and the effectiveness is verified via theoretical analysis and numerical simulations. The controller design is irrelevant to concrete system structure and initial values. It has strong robustness and broad application perspective.
Chaotic behaviour of a predator-prey system
Kooi, B.W.; Boer, M.P.
2003-01-01
Generally a predator-prey system is modelled by two ordinary differential equations which describe the rate of changes of the biomasses. Since such a system is two-dimensional no chaotic behaviour can occur. In the popular Rosenzweig-MacArthur model, which replaced the Lotka-Volterra model, a stable
Learning to simulate and predict chaotic dynamical systems
Bakker, R.
2007-01-01
With precise knowledge of the rules which govern a deterministic chaotic system, it is possible to interact with the system and change its dynamics. This research is part of a larger project, in which chaos control is used to improve the bubbling behavior of multi-phase chemical reactors. Chaos con
Experimental chaotic quantification in bistable vortex induced vibration systems
Huynh, B. H.; Tjahjowidodo, T.
2017-02-01
The study of energy harvesting by means of vortex induced vibration systems has been initiated a few years ago and it is considered to be potential as a low water current energy source. The energy harvester is realized by exposing an elastically supported blunt structure under water flow. However, it is realized that the system will only perform at a limited operating range (water flow) that is attributed to the resonance phenomenon that occurs only at a frequency that corresponds to the fluid flow. An introduction of nonlinear elements seems to be a prominent solution to overcome the problem. Among many nonlinear elements, a bistable spring is known to be able to improve the harvested power by a vortex induced vibrations (VIV) based energy converter at the low velocity water flows. However, it is also observed that chaotic vibrations will occur at different operating ranges that will erratically diminish the harvested power and cause a difficulty in controlling the system that is due to the unpredictability in motions of the VIV structure. In order to design a bistable VIV energy converter with improved harvested power and minimum negative effect of chaotic vibrations, the bifurcation map of the system for varying governing parameters is highly on demand. In this study, chaotic vibrations of a VIV energy converter enhanced by a bistable stiffness element are quantified in a wide range of the governing parameters, i.e. damping and bistable gap. Chaotic vibrations of the bistable VIV energy converter are simulated by utilization of a wake oscillator model and quantified based on the calculation of the Lyapunov exponent. Ultimately, a series of experiments of the system in a water tunnel, facilitated by a computer-based force-feedback testing platform, is carried out to validate the existence of chaotic responses. The main challenge in dealing with experimental data is in distinguishing chaotic response from noise-contaminated periodic responses as noise will smear
Groot, L.F.M.
The purpose of this paper is twofold. First, it exhibits that standard tools in the measurement of income inequality, such as the Lorenz curve and the Gini-index, can successfully be applied to the issues of inequality measurement of carbon emissions and the equity of abatement policies across
Video Encryption Based on Chaotic Systems in the Compression Domain
Directory of Open Access Journals (Sweden)
Ali Abdulgader
2012-01-01
Full Text Available With the development of the internet and multimedia technology digital video encryption has attracted a great deal of research interest in the recent few years in applications. In this paper, we propose a method to encrypt video data. The proposed algorithm is based on the MPEG video coding standard. It selectively encrypts some DCT coefficients in the I frame, B frame and P frame in MPEG video compression by using chaotic systems. The key in this paper is chaotic sequence based on logistic mapping. It can produce the pseudo-random sequences with good randomness. The experimental results based on chaotic maps prove the effectiveness of the proposed method, showing advantages of large key space and high-level security. The proposed algorithm was measured through a series of tests and achieved good results. The results indicate that the algorithm can be implemented for video encryption efficiently and it provides considerable levels of security.
Fully Digital Chaotic Differential Equation-based Systems And Methods
Radwan, Ahmed Gomaa Ahmed
2012-09-06
Various embodiments are provided for fully digital chaotic differential equation-based systems and methods. In one embodiment, among others, a digital circuit includes digital state registers and one or more digital logic modules configured to obtain a first value from two or more of the digital state registers; determine a second value based upon the obtained first values and a chaotic differential equation; and provide the second value to set a state of one of the plurality of digital state registers. In another embodiment, a digital circuit includes digital state registers, digital logic modules configured to obtain outputs from a subset of the digital shift registers and to provide the input based upon a chaotic differential equation for setting a state of at least one of the subset of digital shift registers, and a digital clock configured to provide a clock signal for operating the digital shift registers.
Optimal periodic orbits of continuous time chaotic systems
Yang; Hunt; Ott
2000-08-01
In previous work [B. R. Hunt and E. Ott, Phys. Rev. Lett. 76, 2254 (1996); Phys. Rev. E 54, 328, (1996)], based on numerical experiments and analysis, it was conjectured that the optimal orbit selected from all possible orbits on a chaotic attractor is "typically" a periodic orbit of low period. By an optimal orbit we mean the orbit that yields the largest value of a time average of a given smooth "performance" function of the system state. Thus optimality is defined with respect to the given performance function. (The study of optimal orbits is of interest in at least three contexts: controlling chaos, embedding of low-dimensional attractors of high-dimensional dynamical systems in low-dimensional measurement spaces, and bubbling bifurcations of synchronized chaotic systems.) Here we extend this previous work. In particular, the previous work was for discrete time dynamical systems, and here we shall consider continuous time systems (flows). An essential difference for flows is that chaotic attractors can have embedded within them, not only unstable periodic orbits, but also unstable steady states, and we find that optimality can often occur on steady states. We also shed further light on the sense in which optimality is "typically" achieved at low period. In particular, we find that, as a system parameter is tuned to be closer to a crisis of the chaotic attractor, optimality may occur at higher period.
State Prediction of Chaotic System Based on ANN Model
Institute of Scientific and Technical Information of China (English)
YUE Yi-hong; HAN Wen-xiu
2002-01-01
The choice of time delay and embedding dimension is very important to the phase space reconstruction of any chaotic time series. In this paper, we determine optimal time delay by computing autocorrelation function of time series. Optimal embedding dimension is given by means of the relation between embedding dimension and correlation dimension of chaotic time series. Based on the methods above,we choose ANN model to appoximate the given true system. At the same time, a new algorithm is applied to determine the network weights. At the end of this paper, the theory above is demonstrated through the research of time series generated by Logistic map.
Localized Structures Embedded in the Eigenfunctions of Chaotic Hamiltonian Systems
Vergini, E G
1998-01-01
We study quantum localization phenomena in chaotic systems with a parameter. The parametric motion of energy levels proceeds without crossing any other and the defined avoided crossings quantify the interaction between states. We propose the elimination of avoided crossings as the natural mechanism to uncover localized structures. We describe an efficient method for the elimination of avoided crossings in chaotic billiards and apply it to the stadium billiard. We find many scars of short periodic orbits revealing the skeleton on which quantum mechanics is built. Moreover, we have observed strong interaction between similar localized structures.
Lorenz Wind Disturbance Model Based on Grey Generated Components
Yagang Zhang; Jingyun Yang; Kangcheng Wang; Yinding Wang
2014-01-01
In order to meet the needs of wind speed prediction in wind farms, we consider the influence of random atmospheric disturbances on wind variations. Considering a simplified fluid convection mode, a Lorenz system can be employed as an atmospheric disturbance model. Here Lorenz disturbance is defined as the European norm of the solutions of the Lorenz equation. Grey generating and accumulated generating models are employed to explore the relationship between wind speed and its related disturban...
Indian Academy of Sciences (India)
Mayank Srivastava; Saurabh K Agrawal; Subir Das
2013-09-01
The article deals with adaptive projective synchronization between two different chaotic systems with parametric uncertainties and external disturbances. Based on Lyapunov stability theory, the projective synchronization between a pair of different chaotic systems with fully unknown parameters are derived. An adaptive control law and a parameter update rule for uncertain parameters are designed such that the chaotic response system controls the chaotic drive system. Numerical simulation results are performed to explain the effectiveness and feasibility of the techniques.
Evidence for Lorenz-type chaos in a laser
Weiss, C. O.; Brock, J.
1986-12-01
Observations of the dynamics of a single-mode, traveling-wave laser under bad-cavity conditions are reported. The sequence of instabilities occurring on resonator tuning corresponds in detail to the transition to chaos of the logistic equation. Period-doubling cascade, reverse ('noisy') cascade, and the regular period-3 and -5 windows in the chaotic range are observed. At presumably homogeneous broadening conditions the transition from CW to chaotic emission is abrupt on pump variation. All of the observed features including instability pump thresholds and characteristics of the chaotic laser pulses agree with predictions of the Lorenz equations.
An Adaptive Tracking Control of Fractional-Order Chaotic Systems with Uncertain System Parameter
Directory of Open Access Journals (Sweden)
Ping Zhou
2011-01-01
Full Text Available An adaptive tracking control scheme is presented for fractional-order chaotic systems with uncertain parameter. It is theoretically proved that this approach can make the uncertain parameter fractional-order chaotic system track any given reference signal and the uncertain system parameter is estimated through the adaptive tracking control process. Furthermore, the reference signal may belong to other integer-orders chaotic system or belong to different fractional-order chaotic system with different fractional orders. Two examples are presented to demonstrate the effectiveness of the proposed method.
Dynamical Analysis of the Lorenz-84 Atmospheric Circulation Model
Directory of Open Access Journals (Sweden)
Hu Wang
2014-01-01
Full Text Available The dynamical behaviors of the Lorenz-84 atmospheric circulation model are investigated based on qualitative theory and numerical simulations. The stability and local bifurcation conditions of the Lorenz-84 atmospheric circulation model are obtained. It is also shown that when the bifurcation parameter exceeds a critical value, the Hopf bifurcation occurs in this model. Then, the conditions of the supercritical and subcritical bifurcation are derived through the normal form theory. Finally, the chaotic behavior of the model is also discussed, the bifurcation diagrams and Lyapunov exponents spectrum for the corresponding parameter are obtained, and the parameter interval ranges of limit cycle and chaotic attractor are calculated in further. Especially, a computer-assisted proof of the chaoticity of the model is presented by a topological horseshoe theory.
Multiscality in the Dynamics of Coupled Chaotic Systems
DEFF Research Database (Denmark)
Pavlov, A.N.; Sosnovtseva, Olga; Ziganshin, A.R.
2002-01-01
We investigate the scaling features of complex motions in systems of two coupled chaotic oscillators by means of the wavelet-transform modulus maxima method and the detrended fluctuation analysis. We show that the transition from asynchronous to synchronous dynamics typically reduces the degree...
Chaotic Dynamics and Transport in Classical and Quantum Systems
Energy Technology Data Exchange (ETDEWEB)
NONE
2003-07-01
The aim of this summer school is to provide a set of extended and pedagogical lectures, on the major present-day topics in dynamical systems and statistical mechanics including applications. Some articles are dedicated to chaotic transport in plasma turbulence and to quantum chaos. This document gathers the summaries of some presentations.
Intrinsic Chaoticity in Stable Classical Systems and Quantum Fluctuations
De Martino, S; Illuminati, F
1997-01-01
We postulate the existence of a universal Keplerian tremor for any stable classical complex system on every scale. Deriving the characteristic unit of action $\\alpha$ for each classical interaction, we obtain in all cases $\\alpha connected to an intrinsic chaoticity needed to assure stability of matter. Introducing temperature, we provide further consistency checks corroborating our hypothesis.
Synchronization in driven chaotic systems: Diagnostics and bifurcations
DEFF Research Database (Denmark)
Vadivasova, T.E.; Balanov, A.G.; Sosnovtseva, O.V.;
1999-01-01
We investigate generic aspects of chaos synchronization in an externally forced Rössler system. By comparing different diagnostic methods, we show the existence of a well-defined cut-off of synchronization associated with the transition from weak to fully developed chaos. Two types of chaotic beh...... behavior, differing by the number of vanishing Lyapunov exponents, are observed outside the synchronization regime....
Tracking Control and Synchronization for Two-Dimension Discrete Chaotic Systems
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
The popular method of tracking control and synchronization for two-dimension discrete chaotic systems is put forward in this paper, and the chaotic system track arbitrarily reference signal is realized. This method is applied to two chaotic systems, and one can get good control result.
Directory of Open Access Journals (Sweden)
Ricardo Aguilar-López
2016-01-01
Full Text Available This paper proposes a synchronization methodology of two chaotic oscillators under the framework of identical synchronization and master-slave configuration. The proposed methodology is based on state observer design under the frame of control theory; the observer structure provides finite-time synchronization convergence by cancelling the upper bounds of the main nonlinearities of the chaotic oscillator. The above is showed via an analysis of the dynamic of the so called synchronization error. Numerical experiments corroborate the satisfactory results of the proposed scheme.
Aguilar-López, Ricardo
2016-01-01
This paper proposes a synchronization methodology of two chaotic oscillators under the framework of identical synchronization and master-slave configuration. The proposed methodology is based on state observer design under the frame of control theory; the observer structure provides finite-time synchronization convergence by cancelling the upper bounds of the main nonlinearities of the chaotic oscillator. The above is showed via an analysis of the dynamic of the so called synchronization error. Numerical experiments corroborate the satisfactory results of the proposed scheme. PMID:27738651
Linearly Coupled Synchronization of the New Chaotic Systems
Institute of Scientific and Technical Information of China (English)
LU Jun-an; ZHOU Jin; LI Yi-tian
2005-01-01
This paper investigates synchronization within the new systems, which we denote as Liu system in this paper. New stability criteria for synchronization of linearly coupled Liu systems are attained using the Lyapunov method. Some sufficient conditions for synchronization are concluded through rigorous mathematical theory, which can be further applied to more chaotic systems. Moreover, numerical simulations are given to show the effectiveness of our synchronization criterions.
Nonlinear Filtering Preserves Chaotic Synchronization via Master-Slave System
Directory of Open Access Journals (Sweden)
J. S. González-Salas
2013-01-01
Full Text Available We present a study on a class of interconnected nonlinear systems and give some criteria for them to behave like a filter. Some chaotic systems present this kind of interconnected nonlinear structure, which enables the synchronization of a master-slave system. Interconnected nonlinear filters have been defined in terms of interconnected nonlinear systems. Furthermore, their behaviors have been studied numerically and theoretically on different input signals.
National Research Council Canada - National Science Library
Sundarapandian Vaidyanathan
2011-01-01
...).Explicitly, new state feedback control laws regulating the output of the modified Arneodo chaotic systemhave been derived so as to regulate the output of the modified Arneodo chaotic system have been...
Fuzzy Sliding Mode Control for Hyper Chaotic Chen System
Directory of Open Access Journals (Sweden)
SARAILOO, M.
2012-02-01
Full Text Available In this paper, a fuzzy sliding mode control method is proposed for stabilizing hyper chaotic Chen system. The main objective of the control scheme is to stabilize unstable equilibrium point of the system by controlling the states of the system so that they converge to a pre-defined sliding surface and remain on it. A fuzzy control technique is also utilized in order to overcome the main disadvantage of sliding mode control methods, i.e. chattering problem. It is shown that the equilibrium point of the system is stabilized by using the proposed method. A stability analysis is also performed to prove that the states of the system converge to the sliding surface and remain on it. Simulations show that the control method can be effectively applied to Chen system when it performs hyper chaotic behavior.
Chaotic Motion in the Solar System and Beyond
Lissauer, Jack; DeVincenzi, Donald (Technical Monitor)
2001-01-01
The motion of planetary bodies is the archetypal clockwork system. Indeed, clocks and calendars were developed to keep track of the relative motions of the Earth, the Sun and the Moon. However, studies over the past few decades imply that this predictable regularity does not extend to small bodies, nor does it apply to the precise trajectories of the planets themselves over long timescale.s. Various examples of chaotic motion within our Solar System and, extrasolar planetary systems will be discussed.
Energy Technology Data Exchange (ETDEWEB)
Groot, L. [Utrecht University, Utrecht School of Economics, Janskerkhof 12, 3512 BL Utrecht (Netherlands)
2008-11-15
The purpose of this paper is twofold. First, it exhibits that standard tools in the measurement of income inequality, such as the Lorenz curve and the Gini-index, can successfully be applied to the issues of inequality measurement of carbon emissions and the equity of abatement policies across countries. These tools allow policy-makers and the general public to grasp at a single glance the impact of conventional distribution rules such as equal caps or grandfathering, or more sophisticated ones, on the distribution of greenhouse gas emissions. Second, using the Samuelson rule for the optimal provision of a public good, the Pareto-optimal distribution of carbon emissions is compared with the distribution that follows if countries follow Nash-Cournot abatement strategies. It is shown that the Pareto-optimal distribution under the Samuelson rule can be approximated by the equal cap division, represented by the diagonal in the Lorenz curve diagram.
Letellier, Christophe; Amaral, Gleison F. V.; Aguirre, Luis A.
2007-06-01
The characterization of chaotic attractors has been a widely addressed problem and there are now many different techniques to define their nature in a rather accurate way, at least in the case of a three-dimensional system. Nevertheless, the link between the structure of the ordinary differential equations and the topology of their solutions is still missing and only a few necessary conditions on the algebraic structure are known today. By using a feedback circuit analysis, it is shown that it is possible to identify the relevant terms of the equations, that is, the terms that really contribute to the structure of the phase portrait. Such analysis also provides some guidelines for constructing piecewise affine models. Moreover, equivalence classes can be defined on the basis of the active feedback circuits involved.
On nonlinear control design for autonomous chaotic systems of integer and fractional orders
Energy Technology Data Exchange (ETDEWEB)
Ahmad, Wajdi M. E-mail: wajdi@sharjah.ac.ae; Harb, Ahmad M. E-mail: aharb@just.edu.jo
2003-11-01
In this paper, we address the problem of chaos control for autonomous nonlinear chaotic systems. We use the recursive 'backstepping' method of nonlinear control design to derive the nonlinear controllers. The controller effect is to stabilize the output chaotic trajectory by driving it to the nearest equilibrium point in the basin of attraction. We study two nonlinear chaotic systems: an electronic chaotic oscillator model, and a mechanical chaotic 'jerk' model. We demonstrate the robustness of the derived controllers against system order reduction arising from the use of fractional integrators in the system models. Our results are validated via numerical simulations.
Loss of lag synchronization in coupled chaotic systems
Sosnovtseva, O. V.; Balanov, A. G.; Vadivasova, T. E.; Astakhov, V. V.; Mosekilde, Erik
1999-01-01
Lag synchronization denotes a particular form of synchronization in which the amplitudes of two interacting, nonidentical chaotic oscillators are correlated but there is a characteristic time delay between them. We study transitions to and between different forms of synchronization for the attractors defined as "in-phase" and "out-of-phase" and investigate the processes by which lag synchronization is lost in two coupled Rossler systems. With a small frequency mismatch between the two systems...
An exponential polynomial observer for synchronization of chaotic systems
Mata-Machuca, J. L.; Martínez-Guerra, R.; Aguilar-López, R.
2010-12-01
In this paper, we consider the synchronization problem via nonlinear observer design. A new exponential polynomial observer for a class of nonlinear oscillators is proposed, which is robust against output noises. A sufficient condition for synchronization is derived analytically with the help of Lyapunov stability theory. The proposed technique has been applied to synchronize chaotic systems (Rikitake and Rössler systems) by means of numerical simulation.
Intelligent Controller for Synchronization New Three Dimensional Chaotic System
Directory of Open Access Journals (Sweden)
Alireza Sahab
2014-07-01
Full Text Available One of the most important phenomena of some systems is chaos which is caused by nonlinear dynamics. In this paper, the new 3 dimensional chaotic system is first investigated and then utilized an intelligent controller based on brain emotional learning (BELBIC, this new chaotic system is synchronized. The BELBIC consists of reward signal which accepts positive values. Improper selection of the parameters causes an improper behavior which may cause serious problems such as instability of the system. It is needed to optimize these parameters. Genetic Algorithm (GA, Cuckoo Optimization Algorithm (COA, Particle Swarm Optimization Algorithm (PSO and Imperialist Competitive Algorithm (ICA are used to compute the optimal parameters for the reward signal of BELBIC. These algorithms can select appropriate and optimal values for the parameters. These minimize the Cost Function, so the optimal values for the parameters will be founded. Selected cost function is defined to minimizing the least square errors. Cost function enforces the system errors to decay to zero rapidly. Numerical simulation will show that this method much better, faster and more effective than previous methods can be new 3D chaotic system mode to bring synchronized.
A Retrospection of Chaotic Phenomena in Electrical Systems
Directory of Open Access Journals (Sweden)
Umesh Kumar
1998-01-01
Full Text Available In the last decade new phenomena have been observed in all areas of non linear dynamics, principal among these being ‘Chaotic phenomena’. Chaos has been reported virtually from every scientific discipline. This paper summarizes a study of the chaotic phenomena in electrical systems and attempts to translate the mathematical ideas and techniques into language that engineers and applied scientists can use to study ‘Chaos’. Towards this end, the paper has summarized the study of chaos in several examples like Chua’s circuit family; Folded Torus circuit; non-autonomous circuits; switched capacitor circuits and hyper-chaos circuits. As observed in power systems, control systems and digital filters, chaos has been exhibited and shown on examples.
基于联合混沌系统的保密通信方法研究%Study for Secure Communications Based on Combined Discrete Chaotic Systems
Institute of Scientific and Technical Information of China (English)
王兆霞; 陈增强; 袁著祉; 郝庭柱; 杨保和
2002-01-01
We presents an approach for secure communication by using combined discrete chaotic systems.Two well-known chaotic systems,Henon and Lorenz are combined as an illustrative example to demonstrate the effectiveness of the proposed approach.The chaotic systems adopt backstepping design an dead-beat synchronization scheme because of the characteristics that can reach to synchronization quickly.Simple chaotic masking technique is used in the simulation.The results as well as the theory analysis show that the proposed method is feasible and is of more security performance compared to the non-combined chaotic systems.The proposed scheme is not only can be carried out easily and easy to control,but also advances the security performance of whole system.%提出了利用离散混沌系统联合加密来实现保密通信的方法.引用了著名的Henon和Lorenz混沌系统作为仿真实验证明了其可行性,混沌系统均采用backstepping和dead-beat同步设计方案,以快速达到同步.理论分析证明,此种方法不仅是可行的,且与非联合混沌系统相比,加强了保密性.混沌屏蔽技术仿真结果也证明,该系统不仅实现简单,容易控制,而且增强了整个系统的保密性能.
Chaotic time series. Part II. System Identification and Prediction
Directory of Open Access Journals (Sweden)
Bjørn Lillekjendlie
1994-10-01
Full Text Available This paper is the second in a series of two, and describes the current state of the art in modeling and prediction of chaotic time series. Sample data from deterministic non-linear systems may look stochastic when analysed with linear methods. However, the deterministic structure may be uncovered and non-linear models constructed that allow improved prediction. We give the background for such methods from a geometrical point of view, and briefly describe the following types of methods: global polynomials, local polynomials, multilayer perceptrons and semi-local methods including radial basis functions. Some illustrative examples from known chaotic systems are presented, emphasising the increase in prediction error with time. We compare some of the algorithms with respect to prediction accuracy and storage requirements, and list applications of these methods to real data from widely different areas.
Chaotic time series; 2, system identification and prediction
Lillekjendlie, B
1994-01-01
This paper is the second in a series of two, and describes the current state of the art in modelling and prediction of chaotic time series. Sampled data from deterministic non-linear systems may look stochastic when analysed with linear methods. However, the deterministic structure may be uncovered and non-linear models constructed that allow improved prediction. We give the background for such methods from a geometrical point of view, and briefly describe the following types of methods: global polynomials, local polynomials, multi layer perceptrons and semi-local methods including radial basis functions. Some illustrative examples from known chaotic systems are presented, emphasising the increase in prediction error with time. We compare some of the algorithms with respect to prediction accuracy and storage requirements, and list applications of these methods to real data from widely different areas.
Complete chaotic synchronization in mutually coupled time-delay systems.
Landsman, Alexandra S; Schwartz, Ira B
2007-02-01
Complete chaotic synchronization of end lasers has been observed in a line of mutually coupled, time-delayed system of three lasers, with no direct communication between the end lasers. The present paper uses ideas from generalized synchronization to explain the complete synchronization in the presence of long coupling delays, applied to a model of mutually coupled semiconductor lasers in a line. These ideas significantly simplify the analysis by casting the stability in terms of the local dynamics of each laser. The variational equations near the synchronization manifold are analyzed, and used to derive the synchronization condition that is a function of parameters. The results explain and predict the dependence of synchronization on various parameters, such as time delays, strength of coupling and dissipation. The ideas can be applied to understand complete synchronization in other chaotic systems with coupling delays and no direct communication between synchronized subsystems.
Unstable dimension variability and synchronization of chaotic systems
Viana, R L; Viana, Ricardo L.; Grebogi, Celso
1999-01-01
An aspect of the synchronization dynamics is investigated in this work. We argue analytically and confirm numerically that the chaotic dynamics on the synchronization manifold exhibits unstable dimension variability. Unstable dimension variability is a cause of severe modeling difficulty for physical phenomena, since trajectories obtained from the mathematical model may not be related to trajectories of the actual system. We present and example of unstable dimension variability occurring in a system of two coupled chaotic maps, considering the dynamics in the synchronization manifold and its corresponding transversal direction, where a tongue-like structure is formed. The unstable dimension variability is revealed in the statistical distribution of the finite-time transversal Lyapunov exponent, having both negative and positive values.
A Devaney Chaotic System Which Is Neither Distributively nor Topologically Chaotic
Institute of Scientific and Technical Information of China (English)
Chen Zhi-zhi; Liao Li; Wang Wei
2013-01-01
Weiss proved that Devaney chaos does not imply topological chaos and Oprocha pointed out that Devaney chaos does not imply distributional chaos. In this paper, by constructing a simple example which is Devaney chaotic but neither distributively nor topologically chaotic, we give a unified proof for the results of Weiss and Oprocha.
Dynamics of two coupled chaotic systems driven by external signals
Mancini, H.L. (Héctor Luis); Vidal, G.
2011-01-01
Setting-up a controlled or synchronized state in a space-time chaotic structure targeting an unstable periodic orbit is a key feature of many problems in high dimensional physical, electronics, biological and ecological systems (among others). Formerly, we have shown numerically and experimentally that phase synchronization [M.G. Rosenblum, A.S. Pikovsky, J. Kurths, Phys. Rev. Lett. 78, 4193 (1997)] can be achieved in time dependent hydrodynamic flows [D. Maza, A. Vallone, H.L. Mancini, S....
Synchronization in a unified fractional-order chaotic system
Institute of Scientific and Technical Information of China (English)
Wu Zheng-Mao; Xie Jian-Ying
2007-01-01
In this paper, the synchronization in a unified fractional-order chaotic system is investigated by two methods. One is the frequency-domain method that is analysed by using the Laplace transform theory. The other is the time-domain method that is analysed by using the Lyapunov stability theory. Finally, the numerical simulations are used-to illustrate the effectiveness of the proposed synchronization methods.
Synchronization of a chaotic optical system using control
Lai, Ying-Cheng; Grebogi, Celso
1993-11-01
It has been demonstrated that two identical chaotic systems can be made to synchronize by applying small, judiciously chosen, temporal parameter perturbations to one of them [Y. C. Lai and C. Grebogi, Phys. Rev. E 47, 2357(1993)]. This idea is applied to a nonlinear optical ring resonator modeled by the Ikeda-Hammel-Jones-Maloney map. The average time to achieve synchronization and the effect of noise are also discussed.
Preference of Chaotic Synchronization in a Coupled Laser System
Institute of Scientific and Technical Information of China (English)
ZHOU Yun; ZHU Shi-Qun; WU Liang
2005-01-01
In a coupled laser system, the dynamics of the receiving laser is investigated when two separate transmitting lasers are injected into the receiving laser with different coupling strengths. It is shown that the phenomenon of preference of chaotic synchronization appears under appropriate coupling conditions. The receiving laser will entrain the pulses of either one or both transmitting lasers when the coupling is strong while it keeps its own dynamics when the coupling is weak.
Generation and control of multi-scroll chaotic attractors in fractional order systems
Energy Technology Data Exchange (ETDEWEB)
Ahmad, Wajdi M. [Department of Electrical and Computer Engineering, University of Sharjah, P.O. Box 27272, Sharjah (United Arab Emirates)] e-mail: wajdi@sharjah.ac.ae
2005-08-01
The objective of this paper is twofold: on one hand we demonstrate the generation of multi-scroll attractors in fractional order chaotic systems. Then, we design state feedback controllers to eliminate chaos from the system trajectories. It is demonstrated that modifying the underlying nonlinearity of the fractional chaotic system results in the birth of multiple chaotic attractors, thus forming the so called multi-scroll attractors. The presence of chaotic behavior is evidenced by a positive largest Lyapunov exponent computed for the output time series. We investigate generation and control of multi-scroll attractors in two different models, both of which are fractional order and chaotic: an electronic oscillator, and a mechanical 'jerk' model. The current findings extend previously reported results on generation of n-scroll attractors from the domain of integer order to the domain of fractional order chaotic systems, and addresses the issue of controlling such chaotic behaviors. Our investigations are validated through numerical simulations.
Security analysis of chaotic communication systems based on Volterra-Wiener-Korenberg model
Energy Technology Data Exchange (ETDEWEB)
Lei Min [State Key Lab of Vibration, Shock and Noise, Shanghai Jiao Tong University, Shanghai 200030 (China)] e-mail: leimin@sjtu.edu.cn; Meng Guang [State Key Lab of Vibration, Shock and Noise, Shanghai Jiao Tong University, Shanghai 200030 (China); Feng Zhengjin [Institute of Mechatronic Control System, Shanghai Jiao Tong University, Shanghai 200030 (China)
2006-04-01
Pseudo-randomicity is an important cryptological characteristic for proof of encryption algorithms. This paper proposes a nonlinear detecting method based on Volterra-Wiener-Korenberg model and suggests an autocorrelation function to analyze the pseudo-randomicity of chaotic secure systems under different sampling interval. The results show that: (1) the increase of the order of the chaotic transmitter will not necessarily result in a high degree of security; (2) chaotic secure systems have higher and stronger pseudo-randomicity at sparse sampling interval due to the similarity of chaotic time series to the noise; (3) Volterra-Wiener-Korenberg method can also give a further appropriate sparse sampling interval for improving the security of chaotic secure communication systems. For unmasking chaotic communication systems, the Volterra-Wiener-Korenberg technique can be applied to analyze the chaotic time series with surrogate data.
A novel 3-D jerk chaotic system with three quadratic nonlinearities and its adaptive control
Directory of Open Access Journals (Sweden)
Vaidyanathan Sundarapandian
2016-03-01
Full Text Available This paper announces an eight-term novel 3-D jerk chaotic system with three quadratic nonlinearities. The phase portraits of the novel jerk chaotic system are displayed and the qualitative properties of the jerk system are described. The novel jerk chaotic system has two equilibrium points, which are saddle-foci and unstable. The Lyapunov exponents of the novel jerk chaotic system are obtained as L1 = 0.20572,L2 = 0 and L3 = −1.20824. Since the sum of the Lyapunov exponents of the jerk chaotic system is negative, we conclude that the chaotic system is dissipative. The Kaplan-Yorke dimension of the novel jerk chaotic system is derived as DKY = 2.17026. Next, an adaptive controller is designed via backstepping control method to globally stabilize the novel jerk chaotic system with unknown parameters. Moreover, an adaptive controller is also designed via backstepping control method to achieve global chaos synchronization of the identical jerk chaotic systems with unknown parameters. The backstepping control method is a recursive procedure that links the choice of a Lyapunov function with the design of a controller and guarantees global asymptotic stability of strict feedback systems. MATLAB simulations have been depicted to illustrate the phase portraits of the novel jerk chaotic system and also the adaptive backstepping control results.
A kind of fuzzy control for chaotic systems
Institute of Scientific and Technical Information of China (English)
WANG Hong-wei; MA Guang-fu
2007-01-01
With a T-S fuzzy dynamic model approximating to a non-linear system, the nonlinear system can be decomposed into some local linear models. A variable structure controller based on Lyapunov theories is designed to guarantee the global stability of the T-S fuzzy model. The controlling problems of a nonlinear system can be solved by means of consisting of linear system variable structure control and fuzzy control. The validity of the control method based on the simulating result of two kinds of chaotic systems is shown here.
Linear optimal control of continuous time chaotic systems.
Merat, Kaveh; Abbaszadeh Chekan, Jafar; Salarieh, Hassan; Alasty, Aria
2014-07-01
In this research study, chaos control of continuous time systems has been performed by using dynamic programming technique. In the first step by crossing the response orbits with a selected Poincare section and subsequently applying linear regression method, the continuous time system is converted to a discrete type. Then, by solving the Riccati equation a sub-optimal algorithm has been devised for the obtained discrete chaotic systems. In the next step, by implementing the acquired algorithm on the quantized continuous time system, the chaos has been suppressed in the Rossler and AFM systems as some case studies.
Robust Blind Adaptive Channel Equalization in Chaotic Communication Systems
Institute of Scientific and Technical Information of China (English)
ZHANG Jia-Shu
2006-01-01
Based on the bounded property and statistics of chaotic signal and the idea of set-membership identification,we propose a set-membership generalized least mean square (SM-GLMS) algorithm with variable step size for blind adaptive channel equalization in chaotic communication systems. The steady state performance of the proposed SM-GLMS algorithm is analysed, and comparison with an extended Kalman filter (EKF)-based adaptive algorithm and variable gain least mean square (VG-LMS) algorithm is performed for blind adaptive channel equalization. Simulations show that the proposed SM-GLMS algorithm can provide more significant steady state performance improvement than the EKF-based adaptive algorithm and VG-LMS algorithm.
Quantifying Volume Changing Perturbations in a Wave Chaotic System
Taddese, Biniyam Tesfaye; Moglie, Franco; Antonsen, Thomas M; Ott, Edward; Anlage, Steven M
2012-01-01
A sensor was developed to quantitatively measure perturbations which change the volume of a wave chaotic cavity while leaving its shape intact. The sensors work in the time domain by using either scattering fidelity of the transmitted signals or the classical analog of the Loschmidt echo. The sensors were tested experimentally by inducing volume changing perturbations to a one cubic meter pseudo-integrable, real-world cavity. Perturbations which caused a volume change that is as small as 54 parts in a million were quantitatively measured. These results were obtained by using electromagnetic waves with a wavelength of about $5cm$, therefore, the sensor is sensitive to extreme sub-wavelength changes of the boundaries of a cavity. The experimental results were compared with Finite Difference Time Domain (FDTD) simulation results, and good agreement was found. Furthermore, the sensor was tested using a frequency domain approach on a numerical model of the star graph, which is a representative wave chaotic system....
Gross-Pitaevski map as a chaotic dynamical system
Guarneri, Italo
2017-03-01
The Gross-Pitaevski map is a discrete time, split-operator version of the Gross-Pitaevski dynamics in the circle, for which exponential instability has been recently reported. Here it is studied as a classical dynamical system in its own right. A systematic analysis of Lyapunov exponents exposes strongly chaotic behavior. Exponential growth of energy is then shown to be a direct consequence of rotational invariance and for stationary solutions the full spectrum of Lyapunov exponents is analytically computed. The present analysis includes the "resonant" case, when the free rotation period is commensurate to 2 π , and the map has countably many constants of the motion. Except for lowest-order resonances, this case exhibits an integrable-chaotic transition.
Digital Image Encryption Scheme Based on Multiple Chaotic Systems
Abd El-Latif, Ahmed A.; Li, Li; Zhang, Tiejun; Wang, Ning; Song, Xianhua; Niu, Xiamu
2012-06-01
Image encryption is a challenging task due to the significant level of sophistication achieved by forgerers and other cybercriminals. Advanced encryption methods for secure transmission, storage, and retrieval of digital images are increasingly needed for a number of military, medical, homeland security, and other applications. In this paper, we introduce a new digital image encryption algorithm. The new algorithm employs multiple chaotic systems and cryptographic primitive operations within the encryption process, which are efficiently implemented on modern processors, and adopts round keys for encryption using a chaotic map. Experiments conducted show that the proposed algorithm possesses robust security features such as fairly uniform distribution, high sensitivity to both keys and plainimages, almost ideal entropy, and the ability to highly de-correlate adjacent pixels in the cipherimages. Furthermore, it has a large key space, which greatly increases its security for image encryption applications.
Energy Technology Data Exchange (ETDEWEB)
Liu, Xiaojun [State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi' an Jiaotong University, Xi' an 710049 (China); School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741001 (China); Hong, Ling, E-mail: hongling@mail.xjtu.edu.cn; Jiang, Jun [State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi' an Jiaotong University, Xi' an 710049 (China)
2016-08-15
Global bifurcations include sudden changes in chaotic sets due to crises. There are three types of crises defined by Grebogi et al. [Physica D 7, 181 (1983)]: boundary crisis, interior crisis, and metamorphosis. In this paper, by means of the extended generalized cell mapping (EGCM), boundary and interior crises of a fractional-order Duffing system are studied as one of the system parameters or the fractional derivative order is varied. It is found that a crisis can be generally defined as a collision between a chaotic basic set and a basic set, either periodic or chaotic, to cause a sudden discontinuous change in chaotic sets. Here chaotic sets involve three different kinds: a chaotic attractor, a chaotic saddle on a fractal basin boundary, and a chaotic saddle in the interior of a basin and disjoint from the attractor. A boundary crisis results from the collision of a periodic (or chaotic) attractor with a chaotic (or regular) saddle in the fractal (or smooth) boundary. In such a case, the attractor, together with its basin of attraction, is suddenly destroyed as the control parameter passes through a critical value, leaving behind a chaotic saddle in the place of the original attractor and saddle after the crisis. An interior crisis happens when an unstable chaotic set in the basin of attraction collides with a periodic attractor, which causes the appearance of a new chaotic attractor, while the original attractor and the unstable chaotic set are converted to the part of the chaotic attractor after the crisis. These results further demonstrate that the EGCM is a powerful tool to reveal the mechanism of crises in fractional-order systems.
Liu, Xiaojun; Hong, Ling; Jiang, Jun
2016-08-01
Global bifurcations include sudden changes in chaotic sets due to crises. There are three types of crises defined by Grebogi et al. [Physica D 7, 181 (1983)]: boundary crisis, interior crisis, and metamorphosis. In this paper, by means of the extended generalized cell mapping (EGCM), boundary and interior crises of a fractional-order Duffing system are studied as one of the system parameters or the fractional derivative order is varied. It is found that a crisis can be generally defined as a collision between a chaotic basic set and a basic set, either periodic or chaotic, to cause a sudden discontinuous change in chaotic sets. Here chaotic sets involve three different kinds: a chaotic attractor, a chaotic saddle on a fractal basin boundary, and a chaotic saddle in the interior of a basin and disjoint from the attractor. A boundary crisis results from the collision of a periodic (or chaotic) attractor with a chaotic (or regular) saddle in the fractal (or smooth) boundary. In such a case, the attractor, together with its basin of attraction, is suddenly destroyed as the control parameter passes through a critical value, leaving behind a chaotic saddle in the place of the original attractor and saddle after the crisis. An interior crisis happens when an unstable chaotic set in the basin of attraction collides with a periodic attractor, which causes the appearance of a new chaotic attractor, while the original attractor and the unstable chaotic set are converted to the part of the chaotic attractor after the crisis. These results further demonstrate that the EGCM is a powerful tool to reveal the mechanism of crises in fractional-order systems.
Liu, Xiaojun; Hong, Ling; Jiang, Jun
2016-08-01
Global bifurcations include sudden changes in chaotic sets due to crises. There are three types of crises defined by Grebogi et al. [Physica D 7, 181 (1983)]: boundary crisis, interior crisis, and metamorphosis. In this paper, by means of the extended generalized cell mapping (EGCM), boundary and interior crises of a fractional-order Duffing system are studied as one of the system parameters or the fractional derivative order is varied. It is found that a crisis can be generally defined as a collision between a chaotic basic set and a basic set, either periodic or chaotic, to cause a sudden discontinuous change in chaotic sets. Here chaotic sets involve three different kinds: a chaotic attractor, a chaotic saddle on a fractal basin boundary, and a chaotic saddle in the interior of a basin and disjoint from the attractor. A boundary crisis results from the collision of a periodic (or chaotic) attractor with a chaotic (or regular) saddle in the fractal (or smooth) boundary. In such a case, the attractor, together with its basin of attraction, is suddenly destroyed as the control parameter passes through a critical value, leaving behind a chaotic saddle in the place of the original attractor and saddle after the crisis. An interior crisis happens when an unstable chaotic set in the basin of attraction collides with a periodic attractor, which causes the appearance of a new chaotic attractor, while the original attractor and the unstable chaotic set are converted to the part of the chaotic attractor after the crisis. These results further demonstrate that the EGCM is a powerful tool to reveal the mechanism of crises in fractional-order systems.
A NEW SLIDING MODE CONTROL FOR A CLASS OF UNCERTAIN TIME-DELAY CHAOTIC SYSTEMS
Institute of Scientific and Technical Information of China (English)
LI LI-XIANG; PENG HAI-PENG; GUAN BAO-ZHU; XU JIN-MING
2001-01-01
We propose a new sliding mode control scheme for a class of uncertain time-delay chaotic systems. It is shown that a linear time invariant system with the desired system dynamics is used as a reference model for the output of a time-delay chaotic system to track. A sliding mode controller is then designed to drive the output of the time-delay chaotic system to track the desired linear system. On the sliding mode, the output of the controlled time-delay chaotic system can behave like the desired linear system. A simulation example is given in support of the proposed control scheme.
Analysis and adaptive control of a novel 3-D conservative no-equilibrium chaotic system
Directory of Open Access Journals (Sweden)
Vaidyanathan Sundarapandian
2015-09-01
Full Text Available First, this paper announces a seven-term novel 3-D conservative chaotic system with four quadratic nonlinearities. The conservative chaotic systems are characterized by the important property that they are volume conserving. The phase portraits of the novel conservative chaotic system are displayed and the mathematical properties are discussed. An important property of the proposed novel chaotic system is that it has no equilibrium point. Hence, it displays hidden chaotic attractors. The Lyapunov exponents of the novel conservative chaotic system are obtained as L1 = 0.0395,L2 = 0 and L3 = −0.0395. The Kaplan-Yorke dimension of the novel conservative chaotic system is DKY =3. Next, an adaptive controller is designed to globally stabilize the novel conservative chaotic system with unknown parameters. Moreover, an adaptive controller is also designed to achieve global chaos synchronization of the identical conservative chaotic systems with unknown parameters. MATLAB simulations have been depicted to illustrate the phase portraits of the novel conservative chaotic system and also the adaptive control results.
Gravitational ionization: a chaotic net in the Kepler system
Chicone, C.; Mashhoon, B.; Retzloff, D. G.
1997-03-01
The long-term nonlinear dynamics of a Keplerian binary system under the combined influences of gravitational radiation damping and external tidal perturbations is analysed. Gravitational radiation reaction leads the binary system towards eventual collapse, while the external periodic perturbations could lead to the ionization of the system via Arnold diffusion. When these two opposing tendencies nearly balance each other, interesting chaotic behaviour occurs which is briefly studied in this paper. It is possible to show that periodic orbits can exist in this system for sufficiently small damping. Moreover, we employ the method of averaging to investigate the phenomenon of capture into resonance.
Construction and Verification of a Simple Smooth Chaotic System
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
This article introduces a new chaotic system of three-dimensional quadratic autonomous ordinary differential equations, which can display different attractors with two unstable equilibrium points and four unstable equilibrium points respectively. Dynamical properties of this system are then studied. Furthermore, by applying the undetermined coefficient method, heteroclinic orbit of (S)hil'nikov's type in this system is found and the convergence of the series expansions of this heteroclinic orbit are proved in this article. The (S)hil'nikov's theorem guarantees that this system has Smale horseshoes and the horseshoe chaos.
Impulsive control for synchronization of nonlinear R(o)ssler chaotic systems
Institute of Scientific and Technical Information of China (English)
Li Yang; Liao Xiao-Feng; Li Chuan-Dong; Chen Guo
2006-01-01
This paper reports that an impulsive control theory for synchronization of nonlinear R(o)ssler chaotic systems is developed. A new framework for impulsive synchronization between such chaotic systems is presented, which makes the synchronization error system a linear impulsive control system. Therefore, it is easy to derive the impulsive synchronization law. The proposed impulsive control scheme is illustrated by nonlinear R(o)ssler chaotic systems and the simulation results demonstrate the effectiveness of the method.
Senkerik, Roman; Zelinka, Ivan; Pluhacek, Michal; Davendra, Donald; Oplatková Kominkova, Zuzana
2014-01-01
Evolutionary technique differential evolution (DE) is used for the evolutionary tuning of controller parameters for the stabilization of set of different chaotic systems. The novelty of the approach is that the selected controlled discrete dissipative chaotic system is used also as the chaotic pseudorandom number generator to drive the mutation and crossover process in the DE. The idea was to utilize the hidden chaotic dynamics in pseudorandom sequences given by chaotic map to help differential evolution algorithm search for the best controller settings for the very same chaotic system. The optimizations were performed for three different chaotic systems, two types of case studies and developed cost functions.
Directory of Open Access Journals (Sweden)
Roman Senkerik
2014-01-01
Full Text Available Evolutionary technique differential evolution (DE is used for the evolutionary tuning of controller parameters for the stabilization of set of different chaotic systems. The novelty of the approach is that the selected controlled discrete dissipative chaotic system is used also as the chaotic pseudorandom number generator to drive the mutation and crossover process in the DE. The idea was to utilize the hidden chaotic dynamics in pseudorandom sequences given by chaotic map to help differential evolution algorithm search for the best controller settings for the very same chaotic system. The optimizations were performed for three different chaotic systems, two types of case studies and developed cost functions.
Communications with chaotic optoelectronic systems cryptography and multiplexing
Rontani, Damien
With the rapid development of optical communications and the increasing amount of data exchanged, it has become utterly important to provide effective architectures to protect sensitive data. The use of chaotic optoelectronic devices has already demonstrated great potential in terms of additional computational security at the physical layer of the optical network. However, the determination of the security level and the lack of a multi-user framework are two hurdles which have prevented their deployment on a large scale. In this thesis, we propose to address these two issues. First, we investigate the security of a widely used chaotic generator, the external cavity semiconductor laser (ECSL). This is a time-delay system known for providing complex and high-dimensional chaos, but with a low level of security regarding the identification of its most critical parameter, the time delay. We perform a detailed analysis of the in uence of the ECSL parameters to devise how higher levels of security can be achieved and provide a physical interpretation of their origin. Second, we devise new architectures to multiplex optical chaotic signals and realize multi-user communications at high bit rates. We propose two different approaches exploiting known chaotic optoelectronic devices. The first one uses mutually coupled ECSL and extends typical chaos-based encryption strategies, such as chaos-shift keying (CSK) and chaos modulation (CMo). The second one uses an electro-optical oscillator (EOO) with multiple delayed feedback loops and aims first at transposing coded-division multiple access (CDMA) and then at developing novel strategies of encryption and decryption, when the time-delays of each feedback loop are time-dependent.
A chaotic spread spectrum system for underwater acoustic communication
Ren, Hai-Peng; Bai, Chao; Kong, Qingju; Baptista, Murilo S.; Grebogi, Celso
2017-07-01
Acoustic communication is a key technology to exchange information underwater, which is of great significance to explore marine resources and to marine defense. The underwater acoustic channel is a time-space-frequency varying channel characterized by serious multipath effect, limited frequency band, complex environmental noises and significant Doppler frequency shift phenomenon, which makes underwater acoustic communication with low Bit Error Rate (BER) to be a challenging task. A novel chaotic spread spectrum acoustic communication method with low BER is proposed in this paper. A chaotic signal, generated by a hybrid dynamical system, is used as a spread spectrum sequence at the transmitter end. At the receiver end, a corresponding chaotic matched filter is used to offset the effect of multipath propagation and noise. The proposed method does not require the complicated equalization and modulation-demodulation technologies that are necessary for conventional acoustic communication. Simulation results show that the proposed method has good anti-interference ability and lower BER as compared to other traditional methods.
Chaotic dynamics in N-body systems
Boekholt, Tjarda Coenraad Nico
2015-01-01
Ever since Isaac Newton in 1687 posed the N-body problem, astronomers have been looking for its solutions in order to understand the evolution of dynamical systems, such as our own solar system, star clusters and galaxies. The main difficulty is that small errors grow exponentially, so that numerica
Chaotic dynamics in N-body systems
Boekholt, Tjarda Coenraad Nico
2015-01-01
Ever since Isaac Newton in 1687 posed the N-body problem, astronomers have been looking for its solutions in order to understand the evolution of dynamical systems, such as our own solar system, star clusters and galaxies. The main difficulty is that small errors grow exponentially, so that
A Simple Discrete System with Chaotic Behavior
Asveld, Peter R.J.
1988-01-01
We discuss the behavior of a particular discrete system, viz. Post's system of tag with alphabet $\\{0,1\\}$, deletion number $d=3$, and rules: $0\\rightarrow 00$, $1\\rightarrow 1101$. As initial strings we consider all strings of length less than or equal to 15 as well as all 'worst case' inputs of t
Impulsive control of chaotic systems and its applications in synchronization
Institute of Scientific and Technical Information of China (English)
Wu Bo; Liu Yang; Lu Jian-Quan
2011-01-01
In this paper, some novel sufficient conditions for asymptotic stability of impulsive control systems are presented by comparison systems. The results are used to obtain the conditions under which the chaotic systems can be asymptotically controlled to the origin via impulsive control. Compared with some existing results, our results are more relaxed in the sense that the Lyapunov function is required to be nonincreasing only along a subsequence of switchings. Moreover, a larger upper bound of impulsive intervals for stabilization and synchronization is obtained.
A New Feigenbaum-Like Chaotic 3D System
Directory of Open Access Journals (Sweden)
Huitao Zhao
2014-01-01
Full Text Available Based on Sprott N system, a new three-dimensional autonomous system is reported. It is demonstrated to be chaotic in the sense of having positive largest Lyapunov exponent and fractional dimension. To further understand the complex dynamics of the system, some basic properties such as Lyapunov exponents, bifurcation diagram, Poincaré mapping, and period-doubling route to chaos are analyzed with careful numerical simulations. The obtained results also show that the period-doubling sequence of bifurcations leads to a Feigenbaum-like strange attractor.
Fourier's law for quasi-one-dimensional chaotic quantum systems
Seligman, Thomas H.; Weidenmüller, Hans A.
2011-05-01
We derive Fourier's law for a completely coherent quasi-one-dimensional chaotic quantum system coupled locally to two heat baths at different temperatures. We solve the master equation to first order in the temperature difference. We show that the heat conductance can be expressed as a thermodynamic equilibrium coefficient taken at some intermediate temperature. We use that expression to show that for temperatures large compared to the mean level spacing of the system, the heat conductance is inversely proportional to the level density and, thus, inversely proportional to the length of the system.
Synchronization of Discrete-Time Chaotic Systems in Bandlimited Channels
Directory of Open Access Journals (Sweden)
Marcio Eisencraft
2009-01-01
Full Text Available Over the last couple of decades, many methods for synchronizing chaotic systems have been proposed with communications applications in view. Yet their performance has proved disappointing in face of the nonideal character of usual channels linking transmitter and receiver, that is, due to both noise and signal propagation distortion. Here we consider a discrete-time master-slave system that synchronizes despite channel bandwidth limitations and an allied communication system. Synchronization is achieved introducing a digital filter that limits the spectral content of the feedback loop responsible for producing the transmitted signal.
Adaptive control of uncertain time-delay chaotic systems
Institute of Scientific and Technical Information of China (English)
Zhuhong ZHANG
2005-01-01
This work investigates adaptive control of a large class of uncertain me-delay chaotic systems (UTCSs) with unknown general perturbation terms bounded by a polynomial ( unknown gains). Associated with the different cases of known and unknown system matrices, two corresponding adaptive controllers are proposed to stabilize unstable fixed points of the systems by means of Lyapunov stability theory and linear matrix inequalities (LMI) which can be solved easily by convex optimization algorithms. Two examples are used for examining the effectiveness of the proposed methods.
Liping Chen; Shanbi Wei; Yi Chai; Ranchao Wu
2012-01-01
Projective synchronization between two different fractional-order chaotic systems with fully unknown parameters for drive and response systems is investigated. On the basis of the stability theory of fractional-order differential equations, a suitable and effective adaptive control law and a parameter update rule for unknown parameters are designed, such that projective synchronization between the fractional-order chaotic Chen system and the fractional-order chaotic Lü system with unknown par...
Directory of Open Access Journals (Sweden)
Junwei Sun
2014-01-01
Full Text Available Some important dynamical properties of the memristor chaotic oscillator system have been studied in the paper. A novel hybrid dislocated control method and a general hybrid projective dislocated synchronization scheme have been realized for memristor chaotic oscillator system. The paper firstly presents hybrid dislocated control method for stabilizing chaos to the unstable equilibrium point. Based on the Lyapunov stability theorem, general hybrid projective dislocated synchronization has been studied for the drive memristor chaotic oscillator system and the same response memristor chaotic oscillator system. For the different dimensions, the memristor chaotic oscillator system and the other chaotic system have realized general hybrid projective dislocated synchronization. Numerical simulations are given to show the effectiveness of these methods.
Chaotic motion in nonlinear feedback systems
Energy Technology Data Exchange (ETDEWEB)
Baillieul, J. (Scientific Systems, Inc., Cambridge, MA); Brockett, R.W.; Washburn, R.B.
1980-11-01
New criteria are found which imply the existence of chaos in R/sup n/. These differ significantly from criteria previously reported in the mathematics literature, and in fact our methods apply to a class of systems which do not satisfy the hypotheses of the usual theorems on chaos in R/sup n/. The results are stated in such a way as to preserve the flavor of many well-known frequency-domain stability techniques. The results provide easily verifiable criteria for the existence of chaos in systems which are of dimension greater than one.
Instantaneous frequencies of a chaotic system
Indian Academy of Sciences (India)
C Chandre; T Uzer
2005-03-01
The structure and geometry of high-dimensional, complex dynamical systems is usually hidden under a profusion of numerical data. We show that time–frequency analysis allows one to analyze these data regardless of the number of degrees of freedom. Our method takes snapshots of the system in terms of its instantaneous frequencies defined as ridges of the time–frequency landscape. Using the wavelet transform of a single trajectory, it can characterize key dynamical properties like the extent of chaos, resonance transitions and trappings.
Controlling chaos in a high dimensional system with periodic parametric perturbations
Energy Technology Data Exchange (ETDEWEB)
Mirus, K.A.; Sprott, J.C.
1998-10-01
The effect of applying a periodic perturbation to an accessible parameter of a high-dimensional (coupled-Lorenz) chaotic system is examined. Numerical results indicate that perturbation frequencies near the natural frequencies of the unstable periodic orbits of the chaotic system can result in limit cycles or significantly reduced dimension for relatively small perturbations.
NUMERICAL ANALYSIS OF A CHAOTIC SYSTEM
Institute of Scientific and Technical Information of China (English)
任志坚
2001-01-01
This paper further proves that a single spiral strange attractor can be observed in an extremely simple autonomous electrical circuit by computer simulation. It is of third order and has only one nonlinear element: a three-segment piecewise linear resistor. The digital analyses show that the strange attractor has peculiar features compared with other third-order differential systems.
Chaotic systems in complex phase space
Bender, Carl M; Hook, Daniel W; Weir, David J
2008-01-01
This paper examines numerically the complex classical trajectories of the kicked rotor and the double pendulum. Both of these systems exhibit a transition to chaos, and this feature is studied in complex phase space. Additionally, it is shown that the short-time and long-time behaviors of these two PT-symmetric dynamical models in complex phase space exhibit strong qualitative similarities.
Directory of Open Access Journals (Sweden)
Viet-Thanh Pham
2016-01-01
Full Text Available Discovering systems with hidden attractors is a challenging topic which has received considerable interest of the scientific community recently. This work introduces a new chaotic system having hidden chaotic attractors with an infinite number of equilibrium points. We have studied dynamical properties of such special system via equilibrium analysis, bifurcation diagram, and maximal Lyapunov exponents. In order to confirm the system’s chaotic behavior, the findings of topological horseshoes for the system are presented. In addition, the possibility of synchronization of two new chaotic systems with infinite equilibria is investigated by using adaptive control.
Synchronization of uncertain chaotic systems using active sliding mode control
Energy Technology Data Exchange (ETDEWEB)
Haeri, Mohammad [Advanced Control System Lab, Electrical Engineering Department, Sharif University of Technology, Azadi Avenue, 11365-9363 Tehran (Iran, Islamic Republic of)]. E-mail: haeri@sina.sharif.edu; Tavazoei, Mohammad Saleh [Advanced Control System Lab, Electrical Engineering Department, Sharif University of Technology, Azadi Avenue, 11365-9363 Tehran (Iran, Islamic Republic of); Naseh, Majid Reza [Electrical Engineering Department, Islamic Azad University of Birjand, Birjand (Iran, Islamic Republic of)
2007-08-15
We apply the active sliding mode controller to synchronize two uncertain chaotic systems. Uncertainties are considered both in linear and nonlinear parts of the system dynamics. We have also studied the case that the signals are contaminated by measuring channel noise. It is shown that having some conditions on the uncertainties and noise magnitude, the closed loop stability can be guaranteed. The synchronization errors are shown to be confined into some bounded value. Numerical simulations are presented to evaluate the analysis and effectiveness of the controller.
Loss of lag synchronization in coupled chaotic systems
DEFF Research Database (Denmark)
Sosnovtseva, Olga; Balanov, A G; Vadivasova, T E
1999-01-01
Lag synchronization denotes a particular form of synchronization in which the amplitudes of two interacting, nonidentical chaotic oscillators are correlated but there is a characteristic time delay between them. We study transitions to and between different forms of synchronization...... for the attractors defined as "in-phase" and "out-of-phase" and investigate the processes by which lag synchronization is lost in two coupled Rössler systems. With a small frequency mismatch between the two systems, these processes are related to the occurrence of a peculiar form of basin structure as more and more...
Loss of lag synchronization in coupled chaotic systems
DEFF Research Database (Denmark)
Sosnovtseva, O.V.; Balanov, A.G.; Vadivasova, T.E.
1999-01-01
Lag synchronization denotes a particular form of synchronization in which the amplitudes of two interacting, nonidentical chaotic oscillators are correlated but there is a characteristic time delay between them. We study transitions to and between different forms of synchronization...... for the attractors defined as "in-phase" and "out-of-phase" and investigate the processes by which lag synchronization is lost in two coupled Rossler systems. With a small frequency mismatch between the two systems, these processes are related to the occurrence of a peculiar form of basin structure as more and more...
Chaotic mechanics in systems with impacts and friction
Blazejczyk-Okolewska, Barbara; Kapitaniak, Tomasz; Wojewoda, Jerzy
1999-01-01
This book is devoted to the theory of chaotic oscillations in mechanical systems. Detailed descriptions of the basic types of nonlinearity - impacts and dry friction - are presented. The properties of such behavior are discussed, and the numerical and experimental results obtained by the authors are presented.The dynamic properties of systems described here can be useful in the proper design and use of mechanics where such behavior still creates problems.This book will be very useful for anyone with a fundamental knowledge of nonlinear mechanics who is beginning research in the field.
Global adaptive synchronization of chaotic systems with uncertain parameters
Institute of Scientific and Technical Information of China (English)
李智; 韩崇昭
2002-01-01
We propose a novel adaptive synchronization method for a class of nonlinear chaotic systems with uncertainparameters. Using the chaos control method, we derive a synchronizer, which can make the states of the driven systemglobally track the states of the drive system asymptotically. The advantage of our method is that our problem setting ismore general than those that already exist, and the synchronizer is simply constructed by an analytic formula, withoutknowledge in advance of the unknown bounds of the uncertain parameters. A computer simulation example is given tovalidate the proposed approach.
Chaotic systems in complex phase space
Indian Academy of Sciences (India)
Carl M Bender; Joshua Feinberg; Daniel W Hook; David J Weir
2009-09-01
This paper examines numerically the complex classical trajectories of the kicked rotor and the double pendulum. Both of these systems exhibit a transition to chaos, and this feature is studied in complex phase space. Additionally, it is shown that the short-time and long-time behaviours of these two $\\mathcal{PT}$ -symmetric dynamical models in complex phase space exhibit strong qualitative similarities.
Realization of generalized synchronization between different chaotic systems via scalar controller
Institute of Scientific and Technical Information of China (English)
2007-01-01
In this paper, a very simple generalized synchronization method between different chaotic systems is presented.Only a scalar controller is used in this method. The method of obtaining the scalar controller from chaotic systems is established. The sufficient and necessary condition of generalized synchronization is obtained from a rigorous theory,and the sufficient and necessary condition of generalized synchronization is irrelative to chaotic system itself. Theoretical analyses and simulation results show that the method established in this paper is effective.
Design and Hardware Implementation of a New Chaotic Secure Communication Technique.
Xiong, Li; Lu, Yan-Jun; Zhang, Yong-Fang; Zhang, Xin-Guo; Gupta, Parag
2016-01-01
In this paper, a scheme for chaotic modulation secure communication is proposed based on chaotic synchronization of an improved Lorenz system. For the first time, the intensity limit and stability of the transmitted signal, the characteristics of broadband and the requirements for accuracy of electronic components are presented by Multisim simulation. In addition, some improvements are made on the measurement method and the proposed experimental circuit in order to facilitate the experiments of chaotic synchronization, chaotic non-synchronization, experiment without signal and experiment with signal. To illustrate the effectiveness of the proposed scheme, some numerical simulations are presented. Then, the proposed chaotic secure communication circuit is implemented through analog electronic circuit, which is characterized by its high accuracy and good robustness.
On entanglement spreading in chaotic systems
Mezei, Márk
2016-01-01
We discuss the time dependence of subsystem entropies in interacting quantum systems. As a model for the time dependence, we suggest that the entropy is as large as possible given two constraints: one follows from the existence of an emergent light cone, and the other is a conjecture associated to the "entanglement velocity" $v_E$. We compare this model to new holographic and spin chain computations, and to an operator growth picture. Finally, we introduce a second way of computing the emergent light cone speed in holographic theories that provides a boundary dynamics explanation for a special case of entanglement wedge subregion duality in AdS/CFT.
Chaotic diffusion in the Solar System
Laskar, Jacques
2008-01-01
A statistical analysis is performed over more than 1001 different integrations of the secular equations of the Solar system over 5 Gyr. With this secular system, the probability of the eccentricity of Mercury to reach 0.6 in 5 Gyr is about 1 to 2 %. In order to compare with (Ito and Tanikawa, 2002), we have performed the same analysis without general relativity, and obtained even more orbits of large eccentricity for Mercury. We have performed as well a direct integration of the planetary orbits, without averaging, for a dynamical model that do not include the Moon or general relativity with 10 very close initial conditions over 3 Gyr. The statistics obtained with this reduced set are comparable to the statistics of the secular equations, and in particular we obtain two trajectories for which the eccentricity of Mercury increases beyond 0.8 in less than 1.3 Gyr and 2.8 Gyr respectively. These strong instabilities in the orbital motion of Mecury results from secular resonance beween the perihelion of Jupiter a...
Vaidyanathan, S.; S. Pakiriswamy
2014-01-01
This research work proposes a five-term 3-D novel conservative chaotic system with a quadratic nonlinearity and a quartic nonlinearity. The conservative chaotic systems have the important property that they are volume conserving. The Lyapunov exponents of the 3-D novel chaotic system are obtained as �! = 0.0836, �! = 0 and �! = −0.0836. Since the sum of the Lyapunov exponents is zero, the 3-D novel chaotic system is conservative. Thus, the Kaplan-Yorke dimension of the 3-D novel c...
Hybrid chaotic sequence for QS-CDMA system with RAKE receiver
Institute of Scientific and Technical Information of China (English)
饶妮妮; 许晓晶; 李少谦
2004-01-01
A class of the hybrid chaotic sequences is presented. The generator of the sequences is given and realized by the digital method. The hybrid chaotic sequences exhibit good random properties that are very important for the performance of QS-CDMA system with RAKE receiver. The performance of the system is analyzed when the hybrid chaotic sequences are used as spreading codes in a QS-CDMA system with RAKE receiver and compared with those obtained for m-sequences and logistic sequences. The results show that the hybrid chaotic sequences are a class of very promising spreading codes for QS-CDMA system.
Synchronization and an application of a novel fractional order King Cobra chaotic system.
Muthukumar, P; Balasubramaniam, P; Ratnavelu, K
2014-09-01
In this paper, we design a new three dimensional King Cobra face shaped fractional order chaotic system. The multi-scale synchronization scheme of two fractional order chaotic systems is described. The necessary conditions for the multi-scale synchronization of two identical fractional order King Cobra chaotic systems are derived through feedback control. A new cryptosystem is proposed for an image encryption and decryption by using synchronized fractional order King Cobra chaotic systems with the supports of multiple cryptographic assumptions. The security of the proposed cryptosystem is analyzed by the well known algebraic attacks. Numerical simulations are given to show the effectiveness of the proposed theoretical results.
Simultaneous Synchronization and Anti-Synchronization of Two Identical New 4D Chaotic Systems
Institute of Scientific and Technical Information of China (English)
GUO Rong-Wei
2011-01-01
We investigate the synchronization and anti-synchronization of the new 4D chaotic system and propose a same adaptive controller in the form which not only synchronizes, but also anti-synchronizes two identical new 4D chaotic systems. Numerical simulations verify the correctness and the effectiveness of the proposed theoretical results.%@@ We investigate the synchronization and anti-synchronization of the new 4D chaotic system and propose a same adaptive controller in the form which not only synchronizes, but also anti-synchronizes two identical new 4D chaotic systems.Numerical simulations verify the correctness and the effectiveness of the proposed theoretical results.
Synchronization and an application of a novel fractional order King Cobra chaotic system
Energy Technology Data Exchange (ETDEWEB)
Muthukumar, P., E-mail: muthukumardgl@gmail.com; Balasubramaniam, P., E-mail: balugru@gmail.com [Department of Mathematics, Gandhigram Rural Institute‐Deemed University, Gandhigram 624 302, Tamilnadu (India); Ratnavelu, K., E-mail: kuru052001@gmail.com [Faculty of Science, Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur (Malaysia)
2014-09-01
In this paper, we design a new three dimensional King Cobra face shaped fractional order chaotic system. The multi-scale synchronization scheme of two fractional order chaotic systems is described. The necessary conditions for the multi-scale synchronization of two identical fractional order King Cobra chaotic systems are derived through feedback control. A new cryptosystem is proposed for an image encryption and decryption by using synchronized fractional order King Cobra chaotic systems with the supports of multiple cryptographic assumptions. The security of the proposed cryptosystem is analyzed by the well known algebraic attacks. Numerical simulations are given to show the effectiveness of the proposed theoretical results.
Properties of numerical experiments in chaotic dynamical systems
Yuan, Guo-Cheng
1999-10-01
This dissertation contains four projects that I have worked on during my graduate study at University of Maryland at College Park. These projects are all related to numerical simulations of chaotic dynamical systems. In particular, the two conjectures in Chapter 1 are inspired by the numerical discoveries in Hunt and Ott [1, 2]. In Chapter 2, statistical properties of scalar transport in chaotic flows are investigated by using numerical simulations. In Chapters 3 and 4, I take a different angle and discuss the limitations of numerical simulations; i.e. for certain ``bad'' systems numerical simulations will yield incorrect or at least unreliable results no matter how many digits of precision are used. Chapter 1 discusses the properties of optimal orbits. Given a dynamical system and a function f from the state space to the real numbers, an optimal orbit for f is an orbit over which the average of f is maximal. In this chapter we discuss some basic mathematical aspects of optimal orbits: existence, sensitivity to perturbations of f, and approximability by periodic orbits with low period. For hyperbolic systems, we conjecture that (1)for (topologically) generic smooth functions, there exists an optimal periodic orbit, and (2)the optimal average can be approximated exponentially well by averages over certain periodic orbits with increasing period. In Chapter 2 we theoretically study the power spectrum of passive scalars transported in two dimensional chaotic fluid flows. Using a wave-packet method introduced by Antonsen et al. [3] [4], we numerically investigate several model flows, and confirm that the power spectrum has the k -l- scaling predicted by Batchelor [5]. In Chapter 3 we consider a class of nonhyperbolic systems, for which there are two fixed points in an attractor having a dense trajectory; the unstable manifold of one fixed point has dimension one and the other's is two dimensional. Under the condition that there exists a direction which is more expanding
A chaotic system with a single unstable node
Energy Technology Data Exchange (ETDEWEB)
Sprott, J.C. [Department of Physics, University of Wisconsin, Madison, WI 53706 (United States); Jafari, Sajad, E-mail: sajadjafari@aut.ac.ir [Biomedical Engineering Department, Amirkabir University of Technology, Tehran 15875-4413 (Iran, Islamic Republic of); Pham, Viet-Thanh [School of Electronics and Telecommunications, Hanoi University of Science and Technology, 01 Dai Co Viet, Hanoi (Viet Nam); Hosseini, Zahra Sadat [Biomedical Engineering Department, Amirkabir University of Technology, Tehran 15875-4413 (Iran, Islamic Republic of)
2015-09-25
This paper describes an unusual example of a three-dimensional dissipative chaotic flow with quadratic nonlinearities in which the only equilibrium is an unstable node. The region of parameter space with bounded solutions is relatively small as is the basin of attraction, which accounts for the difficulty of its discovery. Furthermore, for some values of the parameters, the system has an attracting torus, which is uncommon in three-dimensional systems, and this torus can coexist with a strange attractor or with a limit cycle. The limit cycle and strange attractor exhibit symmetry breaking and attractor merging. All the attractors appear to be hidden in that they cannot be found by starting with initial conditions in the vicinity of the equilibrium, and thus they represent a new type of hidden attractor with important and potentially problematic engineering consequences. - Highlights: • An unusual example of a three-dimensional dissipative chaotic flow is introduced. • In this system the only equilibrium is an unstable node. • For some values of the parameters, the system has an attracting torus. • This torus can coexist with a strange attractor or with a limit cycle. • These properties are uncommon in three-dimensional systems.
A chaotic neural network mimicking an olfactory system and its application on image recognition
Institute of Scientific and Technical Information of China (English)
WANG Le; LI Guang; LI Xu; GUO Hong-ji; Walter J. Freeman
2004-01-01
Based on the research of a biological olfactory system, a novel chaotic neural network model - K set model has been established. This chaotic neural network not only simulates the real brain activity of an olfactory system, but also presents a novel chaotic concept for signal processing and pattern recognition. The characteristics of the K set models are investigated and show that a KⅢ model can be used for image pattern classification.
Linear generalized synchronization of chaotic systems with uncertain parameters
Institute of Scientific and Technical Information of China (English)
Jia Zhen
2008-01-01
A more general form of projective synchronization,so called linear generalized synchronization(LGS)is proposed,which includes the generalized projective synchronization(GPS)and the hybrid projective synchronization(HPS)as its special cases.Based on the adaptive technique and Lyapunov stability theory,a general method for achieving the LGS between two chaotic or hyperchaotic systems with uncertain parameters in any scaling matrix is presented.Some numerical simulations are provided to show the effectiveness and feasibility of the proposed synchronization method.
The n-level spectral correlations for chaotic systems
Energy Technology Data Exchange (ETDEWEB)
Nagao, Taro [Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602 (Japan); Mueller, Sebastian [Department of Mathematics, University of Bristol, Bristol BS8 1TW (United Kingdom)
2009-09-18
We study the n-level spectral correlation functions of classically chaotic quantum systems without time-reversal symmetry. According to Bohigas, Giannoni and Schmit's universality conjecture, it is expected that the correlation functions are in agreement with the prediction of the circular unitary ensemble (CUE) of random matrices. A semiclassical resummation formalism allows us to express the correlation functions as sums over pseudo-orbits. Using an extended version of the diagonal approximation on the pseudo-orbit sums, we derive the n-level correlation functions identical to the n x n determinantal correlation functions of the CUE.
Nonlinear time reversal in a wave chaotic system.
Frazier, Matthew; Taddese, Biniyam; Antonsen, Thomas; Anlage, Steven M
2013-02-01
Exploiting the time-reversal invariance and reciprocal properties of the lossless wave equation enables elegantly simple solutions to complex wave-scattering problems and is embodied in the time-reversal mirror. Here we demonstrate the implementation of an electromagnetic time-reversal mirror in a wave chaotic system containing a discrete nonlinearity. We demonstrate that the time-reversed nonlinear excitations reconstruct exclusively upon the source of the nonlinearity. As an example of its utility, we demonstrate a new form of secure communication and point out other applications.
On the recurrence properties of Lorenz'63 model
Gianfelice, Michele; Pelino, Vinicio; Vaienti, Sandro
2011-01-01
Lie-Poisson structure of the Lorenz'63 system gives some physi- cal insight on its dynamical and statistical behaviour considering the evolution of the associated Casimir functions. We study the invariant density and other recurrence features of a Markov expanding Lorenz- like map of the interval arising in the analysis of the predictability of the extreme values reached by particular physical observables evolving in time under the Lorenz'63 dynamics with the classical set of param- eters. This will allow us to further characterize the SRB measure of the system.
A 3D Fractional-Order Chaotic System with Only One Stable Equilibrium and Controlling Chaos
Directory of Open Access Journals (Sweden)
Shiyun Shen
2017-01-01
Full Text Available One 3D fractional-order chaotic system with only one locally asymptotically stable equilibrium is reported. To verify the chaoticity, the maximum Lyapunov exponent (MAXLE with respect to the fractional-order and chaotic attractors are obtained by numerical calculation for this system. Furthermore, by linear scalar controller consisting of a single state variable, one control scheme for stabilization of the 3D fractional-order chaotic system is suggested. The numerical simulations show the feasibility of the control scheme.
Institute of Scientific and Technical Information of China (English)
张家树; 肖先赐; 万继宏
2001-01-01
An adaptive nonlinear feedback-control method is proposed to control continuous-time chaotic dynamical systems,where the adaptive nonlinear controller acts on only one-dimensional error signals between the desired state and the observed chaotic state of a system. The reduced parameter adaptive quadratic predictor used in adaptive feedback cancellation of the nonlinear terms can control the system at any desired state. Computer simulation results on the Lorenz system are shown to demonstrate the effectiveness of this feedback-control method.
Synchronizing chaotic dynamics with uncertainties based on a sliding mode control design.
Yang, Tao; Shao, Hui He
2002-04-01
The synchronization of two chaotic systems with uncertainties is studied in this paper. A feedback controller is provided based on a sliding mode control design. A kind of extended state observer is used to compensate for the systems' uncertainties, such as the structure difference or parameter mismatching, using only the available synchronizing error. Then the feedback controller becomes physically realizable based on the states of the observer, and can be used to synchronize two continuous chaotic systems. Illustrative examples of the synchronization of Duffing and Van der Pol oscillators as well as two Lorenz systems with parameter mismatching are proposed to show the effectiveness of this method.
A mixed analog/digital chaotic neuro-computer system for quadratic assignment problems.
Horio, Yoshihiko; Ikeguchi, Tohru; Aihara, Kazuyuki
2005-01-01
We construct a mixed analog/digital chaotic neuro-computer prototype system for quadratic assignment problems (QAPs). The QAP is one of the difficult NP-hard problems, and includes several real-world applications. Chaotic neural networks have been used to solve combinatorial optimization problems through chaotic search dynamics, which efficiently searches optimal or near optimal solutions. However, preliminary experiments have shown that, although it obtained good feasible solutions, the Hopfield-type chaotic neuro-computer hardware system could not obtain the optimal solution of the QAP. Therefore, in the present study, we improve the system performance by adopting a solution construction method, which constructs a feasible solution using the analog internal state values of the chaotic neurons at each iteration. In order to include the construction method into our hardware, we install a multi-channel analog-to-digital conversion system to observe the internal states of the chaotic neurons. We show experimentally that a great improvement in the system performance over the original Hopfield-type chaotic neuro-computer is obtained. That is, we obtain the optimal solution for the size-10 QAP in less than 1000 iterations. In addition, we propose a guideline for parameter tuning of the chaotic neuro-computer system according to the observation of the internal states of several chaotic neurons in the network.
Controlling a Chaotic System through Control Parameter Self-Modulation
Energy Technology Data Exchange (ETDEWEB)
Pastor, I.
1994-07-01
A method for obtaining active control of a chaotic system based on the modulation of a control parameter by adding to it a small perturbation proportional to one output signal is proposed. From a theoretical point of view, chaos can be stabilized in the framework of this method because small modifications of the vector field controlling the dynamics are allowed, and thus some of the previously oxi sting unstable periodic trajectories can be made stable. The method is much inspired on recent treatments of some related problems, and i t is compared with them. One of its most attractive features is that is should be very easy to implement it on real experiments. The method is tested on a system of ordinary differential equations model ling the coupling of two se If - oscillating electronic circuits (van der Pol oscillators). Some brief comments are made on the no possibility that it could be applied to complex spatio-temporal systems where multiple chaotic structures can coexist for some values of the control parameters. (Author) 28 refs.
Passive control of Permanent Magnet Synchronous Motor chaotic system based on state observer
Institute of Scientific and Technical Information of China (English)
QI Dong-lian; WANG Qiao
2006-01-01
Passive system theory was applied to propose a new passive control method with nonlinear observer of the Permanent Magnet Synchronous Motor chaotic system. Through constructing a Lyapunov function, the subsystem of the Permanent Magnet Synchronous Motor chaotic system could be proved to be globally stable at the equilibrium point. Then a controller with smooth state feedback is designed so that the Permanent Magnet Synchronous Motor chaotic system can be equivalent to a passive system.To get the state variables of the controller, the nonlinear observer is also studied. It is found that the outputs of the nonlinear observer can approximate the state variables of the Permanent Magnet Synchronous Motor chaotic system if the system's nonlinear function is a globally Lipschitz function. Simulation results showed that the equivalent passive system of Permanent Magnet Synchronous Motor chaotic system could be globally asymptotically stabilized by smooth state feedback in the observed parameter convergence condition area.
Global Chaos Synchronization Between Two New DifferentChaotic Systems Via Active Control
Institute of Scientific and Technical Information of China (English)
XU Guang-Li
2009-01-01
This work presents chaos synchronization between two new different chaotic systems by using active control.The proposed controller ensures that the states of the controlled chaotic response system asymptotically synchronizes the states of the drive system.Numerical simulations are shown to verify the result.
Global Chaos Synchronization between Two New Different Chaotic Systems via Active Control
Institute of Scientific and Technical Information of China (English)
SUN Feng-Yun
2006-01-01
We present chaos synchronization between two new different chaotic systems by using active control.The proposed controller ensures that the states of the controlled chaotic response system asymptotically synchronizes the states of the drive system.Numerical simulations are shown to verify the result.
Hierarchical fractal Weyl laws for chaotic resonance states in open mixed systems.
Körber, M J; Michler, M; Bäcker, A; Ketzmerick, R
2013-09-13
In open chaotic systems the number of long-lived resonance states obeys a fractal Weyl law, which depends on the fractal dimension of the chaotic saddle. We study the generic case of a mixed phase space with regular and chaotic dynamics. We find a hierarchy of fractal Weyl laws, one for each region of the hierarchical decomposition of the chaotic phase-space component. This is based on our observation of hierarchical resonance states localizing on these regions. Numerically this is verified for the standard map and a hierarchical model system.
Multiswitching combination–combination synchronization of chaotic systems
Indian Academy of Sciences (India)
AYUB KHAN; DINESH KHATTAR; NITISH PRAJAPATI
2017-03-01
In this paper, a novel synchronization scheme is investigated for a class of chaotic systems. Themultiswitching synchronization scheme is extended to the combination–combination synchronization scheme such that the combination of state variables of two drive systems synchronize with different combination of state variables of two response systems, simultaneously. The new scheme, multiswitching combination–combination synchronization (MSCCS), is a notable extension of the earlier multiswitching schemes concerning only the single drive–response system model. Various multiswitching modified projective synchronization schemes are obtained as special cases of MSCCS, for a suitable choice of scaling factors. Suitable controllers have been designed and using Lyapunov stability theory sufficient condition is obtained to achieve MSCCS between four hyperchaotic systems and the corresponding theoretical proof is given. Numerical simulations are performed to validate the theoretical results.
Chaotic Charged System Search with a Feasible-Based Method for Constraint Optimization Problems
Directory of Open Access Journals (Sweden)
B. Nouhi
2013-01-01
Full Text Available Recently developed chaotic charged system search was combined to feasible-based method to solve constraint engineering optimization problems. Using chaotic maps into the CSS increases the global search mobility for a better global optimization. In the present method, an improved feasible-based method is utilized to handle the constraints. Some constraint design examples are tested using the new chaotic-based methods, and the results are compared to recognize the most efficient and powerful algorithm.
Stochastic perturbations in open chaotic systems: random versus noisy maps.
Bódai, Tamás; Altmann, Eduardo G; Endler, Antonio
2013-04-01
We investigate the effects of random perturbations on fully chaotic open systems. Perturbations can be applied to each trajectory independently (white noise) or simultaneously to all trajectories (random map). We compare these two scenarios by generalizing the theory of open chaotic systems and introducing a time-dependent conditionally-map-invariant measure. For the same perturbation strength we show that the escape rate of the random map is always larger than that of the noisy map. In random maps we show that the escape rate κ and dimensions D of the relevant fractal sets often depend nonmonotonically on the intensity of the random perturbation. We discuss the accuracy (bias) and precision (variance) of finite-size estimators of κ and D, and show that the improvement of the precision of the estimations with the number of trajectories N is extremely slow ([proportionality]1/lnN). We also argue that the finite-size D estimators are typically biased. General theoretical results are combined with analytical calculations and numerical simulations in area-preserving baker maps.
Directory of Open Access Journals (Sweden)
S. Vaidyanathan
2014-11-01
Full Text Available This research work proposes a five-term 3-D novel conservative chaotic system with a quadratic nonlinearity and a quartic nonlinearity. The conservative chaotic systems have the important property that they are volume conserving. The Lyapunov exponents of the 3-D novel chaotic system are obtained as �! = 0.0836, �! = 0 and �! = −0.0836. Since the sum of the Lyapunov exponents is zero, the 3-D novel chaotic system is conservative. Thus, the Kaplan-Yorke dimension of the 3-D novel chaotic system is easily seen as 3.0000. The phase portraits of the novel chaotic system simulated using MATLAB depict the chaotic attractor of the novel system. This research work also discusses other qualitative properties of the system. Next, an adaptive controller is designed to achieve Generalized Projective Synchronization (GPS of two identical novel chaotic systems with unknown system parameters. MATLAB simulations are shown to validate and demonstrate the GPS results derived in this work.
Aguirre, Luis Antonio; Billings, S. A.
This paper investigates the identification of global models from chaotic data corrupted by additive noise. It is verified that noise has a strong influence on the identification of chaotic systems. In particular, there seems to be a critical noise level beyond which the accurate estimation of polynomial models from chaotic data becomes very difficult. Similarities with the estimation of the largest Lyapunov exponent from noisy data suggest that part of the problem might be related to the limited ability of predicting the data records when these are chaotic. A nonlinear filtering scheme is suggested in order to reduce the noise in the data and thereby enable the estimation of good models. This prediction-based filtering incorporates a resetting mechanism which enables the filtering of chaotic data and which is also applicable to non-chaotic data.
Suggested Rules for Designing Secure Communication Systems Utilizing Chaotic Lasers: A Survey
Zhao, Qingchun
2010-01-01
Chaotic communications based on semiconductor lasers have aroused great research interest since 1990s. Physical-layer encryption using chaotic lasers is an alternative to transmit message rapidly and confidentially. There are some practical devices and setups for optical chaotic communications, which are intuitively considered to be secure. However, there is lack of a set of security evaluation rules for these communication setups. According to the recent literature, we summarize several criteria for optical chaotic communications to evaluate the security and point out some methods to enhance the security. These criteria and suggested rules are very helpful in designing secure communication systems using chaotic lasers. Finally we propose some possible hot topics on security analysis of optical chaotic communications in future.
Cross-section fluctuations in chaotic scattering systems
Ericson, Torleif E. O.; Dietz, Barbara; Richter, Achim
2016-10-01
Exact analytical expressions for the cross-section correlation functions of chaotic scattering systems have hitherto been derived only under special conditions. The objective of the present article is to provide expressions that are applicable beyond these restrictions. The derivation is based on a statistical model of Breit-Wigner type for chaotic scattering amplitudes which has been shown to describe the exact analytical results for the scattering (S )-matrix correlation functions accurately. Our results are given in the energy and in the time representations and apply in the whole range from isolated to overlapping resonances. The S -matrix contributions to the cross-section correlations are obtained in terms of explicit irreducible and reducible correlation functions. Consequently, the model can be used for a detailed exploration of the key features of the cross-section correlations and the underlying physical mechanisms. In the region of isolated resonances, the cross-section correlations contain a dominant contribution from the self-correlation term. For narrow states the self-correlations originate predominantly from widely spaced states with exceptionally large partial width. In the asymptotic region of well-overlapping resonances, the cross-section autocorrelation functions are given in terms of the S -matrix autocorrelation functions. For inelastic correlations, in particular, the Ericson fluctuations rapidly dominate in that region. Agreement with known analytical and experimental results is excellent.
High-dimensional chaotic and attractor systems a comprehensive introduction
Ivancevic, Vladimir G
2007-01-01
This is a graduate–level monographic textbook devoted to understanding, prediction and control of high–dimensional chaotic and attractor systems of real life. The objective of the book is to provide the serious reader with a serious scientific tool that will enable the actual performance of competitive research in high–dimensional chaotic and attractor dynamics. The book has nine Chapters. The first Chapter gives a textbook-like introduction into the low-dimensional attractors and chaos. This Chapter has an inspirational character, similar to other books on nonlinear dynamics and deterministic chaos. The second Chapter deals with Smale’s topological transformations of stretching, squeezing and folding (of the system’s phase–space), developed for the purpose of chaos theory. The third Chapter is devoted to Poincaré's 3-body problem and basic techniques of chaos control, mostly of Ott-Grebogi-Yorke type. The fourth Chapter is a review of both Landau’s and topological phase transition theory, as w...
Institute of Scientific and Technical Information of China (English)
Jia Li-Xin; Dai Hao; Hui Meng
2010-01-01
This paper focuses on the synchronisation between fractional-order and integer-order chaotic systems.Based on Lyapunov stability theory and numerical differentiation，a nonlinear feedback controller is obtained to achieve the synchronisation between fractional-order and integer-order chaotic systems.Numerical simulation results are presented to illustrate the effectiveness of this method.
Stabilization of three-dimensional chaotic systems via single state feedback controller
Energy Technology Data Exchange (ETDEWEB)
Yu Wenguang, E-mail: smilewgyu@163.co [School of Statistics and Mathematics, Shandong Economic University, Jinan 250014 (China)
2010-03-29
This Letter investigates the stabilization of three-dimensional chaotic systems, and proposes a novel simple adaptive-feedback controller for chaos control. In comparison with previous methods, the present controller which only contains single state feedback, to our knowledge, is the simplest control scheme for controlling the three-dimensional chaotic system. The results are validated using numerical simulations.
A novel four-wing chaotic attractor generated from a three-dimensional quadratic autonomous system
Institute of Scientific and Technical Information of China (English)
Dong En-Zeng; Chen Zai-Ping; Chen Zeng-Qiang; Yuan Zhu-Zhi
2009-01-01
This paper presents a new 3D quadratic autonomous chaotic system which contains five system parameters and three quadratic cross-product terms, and the system can generate a single four-wing chaotic attractor with wide parameter ranges. Through theoretical analysis, the Hopf bifurcation processes are proved to arise at certain equilibrium points. Numerical bifurcation analysis shows that the system has many interesting complex dynamical behaviours; the system trajectory can evolve to a chaotic attractor from a periodic orbit or a fixed point as the proper parameter varies.Finally, an analog electronic circuit is designed to physically realize the chaotic system; the existence of four-wing chaotic attractor is verified by the analog circuit realization.
Scale-Free Networks Hidden in Chaotic Dynamical Systems
Iba, Takashi
2010-01-01
In this paper, we show our discovery that state-transition networks in several chaotic dynamical systems are "scale-free networks," with a technique to understand a dynamical system as a whole, which we call the analysis for "Discretized-State Transition" (DST) networks; This scale-free nature is found universally in the logistic map, the sine map, the cubic map, the general symmetric map, the sine-circle map, the Gaussian map, and the delayed logistic map. Our findings prove that there is a hidden order in chaos, which has not detected yet. Furthermore, we anticipate that our study opens up a new way to a "network analysis approach to dynamical systems" for understanding complex phenomena.
A new cryptosystem based on spatial chaotic system
Sun, Fuyan; Lü, Zongwang; Liu, Shutang
2010-05-01
Encryption of images is different from that of texts due to some intrinsic features of images such as bulk data capacity and high redundancy, which is generally difficult to handle by traditional methods. This paper proposes a new spatial chaos system (SCS), which is investigated by conducting FIPS 140-1 statistic test, and is especially useful for encryption of digital images. It is shown how to adapt a two dimensional (2D) ergodic matrix obtained from SCS to permute the positions of image pixels and confuse the relationship between the cipher image and plain image simultaneously. Experimental results show that the performance and security of the proposed cryptographic system are better than those of existing lower dimensional chaotic cryptographic systems.
Chaotic oscillator detection system about weak signals in spot welding
Institute of Scientific and Technical Information of China (English)
Kai-lei SONG; Zhen LUO; Feng YE; Xin-xin TANG; Shu-xian YUAN
2009-01-01
Spot welding is an efficient and shortcut processing method used in plate, and its quality detection is very important. However, there are many factors affecting the spot welding quality. Because of the low precision of traditional detection methods, spot welding has seldom been used in the aerospace industry which requires high welding quality. In this article, we give a new weak signal detection model based on chaotic oscillators. Using Melnikov methods and Lyapunov exponent, we can determine the critical values when the system enters in and out of chaos. Through lots of numerical simulations, it can be found that the lowest value of the weak sinusoidal signal the system can detect reach 10-11, and its signal-to-noise ratio (SNR) is = 126 dB. Compared with other detection methods, chaos oscillator detection system not only has a lower threshold value, but also is easy to implement in practice. This model thus has good application prospects.
Directory of Open Access Journals (Sweden)
Li-lian Huang
2013-01-01
Full Text Available The synchronization of nonlinear uncertain chaotic systems is investigated. We propose a sliding mode state observer scheme which combines the sliding mode control with observer theory and apply it into the uncertain chaotic system with unknown parameters and bounded interference. Based on Lyapunov stability theory, the constraints of synchronization and proof are given. This method not only can realize the synchronization of chaotic systems, but also identify the unknown parameters and obtain the correct parameter estimation. Otherwise, the synchronization of chaotic systems with unknown parameters and bounded external disturbances is robust by the design of the sliding surface. Finally, numerical simulations on Liu chaotic system with unknown parameters and disturbances are carried out. Simulation results show that this synchronization and parameter identification has been totally achieved and the effectiveness is verified very well.
Directory of Open Access Journals (Sweden)
S. Vaidyanathan
2013-09-01
Full Text Available This research work describes the modelling of two novel 3-D chaotic systems, the first with a hyperbolic sinusoidal nonlinearity and two quadratic nonlinearities (denoted as system (A and the second with a hyperbolic cosinusoidal nonlinearity and two quadratic nonlinearities (denoted as system (B. In this work, a detailed qualitative analysis of the novel chaotic systems (A and (B has been presented, and the Lyapunov exponents and Kaplan-Yorke dimension of these chaotic systems have been obtained. It is found that the maximal Lyapunov exponent (MLE for the novel chaotic systems (A and (B has a large value, viz. for the system (A and for the system (B. Thus, both the novel chaotic systems (A and (B display strong chaotic behaviour. This research work also discusses the problem of finding adaptive controllers for the global chaos synchronization of identical chaotic systems (A, identical chaotic systems (B and nonidentical chaotic systems (A and (B with unknown system parameters. The adaptive controllers for achieving global chaos synchronization of the novel chaotic systems (A and (B have been derived using adaptive control theory and Lyapunov stability theory. MATLAB simulations have been shown to illustrate the novel chaotic systems (A and (B, and also the adaptive synchronization results derived for the novel chaotic systems (A and (B.
Unfolding of the spectrum for chaotic and mixed systems
Abul-Magd, Ashraf A.; Abul-Magd, Adel Y.
2014-02-01
Random Matrix Theory (RMT) is capable of making predictions for the spectral fluctuations of a physical system only after removing the influence of the level density by unfolding the spectra. When the level density is known, unfolding is done by using the integrated level density to transform the eigenvalues into dimensionless variables with unit mean spacing. When it is not known, as in most practical cases, one usually approximates the level staircase function by a polynomial. We here study the effect of unfolding procedure on the spectral fluctuation of two systems for which the level density is known asymptotically. The first is a time-reversal-invariant chaotic system, which is modeled in RMT by a Gaussian Orthogonal Ensemble (GOE). The second is the case of chaotic systems in which m quantum numbers remain almost undistorted in the early stage of the stochastic transition. The Hamiltonian of a system may be represented by a block diagonal matrix with m blocks of the same size, in which each block is a GOE. Unfolding is done once by using the asymptotic level densities for the eigenvalues of the m blocks and once by representing the integrated level density in terms of polynomials of different orders. We find that the spacing distribution of the eigenvalues shows a little sensitivity to the unfolding method. On the other hand, the variance of level number Σ2(L) is sensitive to the choice of the unfolding function. Unfolding that utilizes low order polynomials enhances Σ2(L) relative to the theoretical value, while the use of high order polynomial reduces it. The optimal value of the order of the unfolding polynomial depends on the dimension of the corresponding ensemble.
Accurate determination of heteroclinic orbits in chaotic dynamical systems
Li, Jizhou; Tomsovic, Steven
2017-03-01
Accurate calculation of heteroclinic and homoclinic orbits can be of significant importance in some classes of dynamical system problems. Yet for very strongly chaotic systems initial deviations from a true orbit will be magnified by a large exponential rate making direct computational methods fail quickly. In this paper, a method is developed that avoids direct calculation of the orbit by making use of the well-known stability property of the invariant unstable and stable manifolds. Under an area-preserving map, this property assures that any initial deviation from the stable (unstable) manifold collapses onto them under inverse (forward) iterations of the map. Using a set of judiciously chosen auxiliary points on the manifolds, long orbit segments can be calculated using the stable and unstable manifold intersections of the heteroclinic (homoclinic) tangle. Detailed calculations using the example of the kicked rotor are provided along with verification of the relation between action differences and certain areas bounded by the manifolds.
Some Chaotic Properties of Discrete Fuzzy Dynamical Systems
Directory of Open Access Journals (Sweden)
Yaoyao Lan
2012-01-01
Full Text Available Letting (X,d be a metric space, f:X→X a continuous map, and (ℱ(X,D the space of nonempty fuzzy compact subsets of X with the Hausdorff metric, one may study the dynamical properties of the Zadeh's extension f̂:ℱ(X→ℱ(X:u↦f̂u. In this paper, we present, as a response to the question proposed by Román-Flores and Chalco-Cano 2008, some chaotic relations between f and f̂. More specifically, we study the transitivity, weakly mixing, periodic density in system (X,f, and its connections with the same ones in its fuzzified system.
Maximal stochastic transport in the Lorenz equations
Energy Technology Data Exchange (ETDEWEB)
Agarwal, Sahil, E-mail: sahil.agarwal@yale.edu [Program in Applied Mathematics, Yale University, New Haven (United States); Wettlaufer, J.S., E-mail: john.wettlaufer@yale.edu [Program in Applied Mathematics, Yale University, New Haven (United States); Departments of Geology & Geophysics, Mathematics and Physics, Yale University, New Haven (United States); Mathematical Institute, University of Oxford, Oxford (United Kingdom); Nordita, Royal Institute of Technology and Stockholm University, Stockholm (Sweden)
2016-01-08
We calculate the stochastic upper bounds for the Lorenz equations using an extension of the background method. In analogy with Rayleigh–Bénard convection the upper bounds are for heat transport versus Rayleigh number. As might be expected, the stochastic upper bounds are larger than the deterministic counterpart of Souza and Doering [1], but their variation with noise amplitude exhibits interesting behavior. Below the transition to chaotic dynamics the upper bounds increase monotonically with noise amplitude. However, in the chaotic regime this monotonicity depends on the number of realizations in the ensemble; at a particular Rayleigh number the bound may increase or decrease with noise amplitude. The origin of this behavior is the coupling between the noise and unstable periodic orbits, the degree of which depends on the degree to which the ensemble represents the ergodic set. This is confirmed by examining the close returns plots of the full solutions to the stochastic equations and the numerical convergence of the noise correlations. The numerical convergence of both the ensemble and time averages of the noise correlations is sufficiently slow that it is the limiting aspect of the realization of these bounds. Finally, we note that the full solutions of the stochastic equations demonstrate that the effect of noise is equivalent to the effect of chaos.
An observer based asymptotic trajectory control using a scalar state for chaotic systems
Institute of Scientific and Technical Information of China (English)
Yu Dong-Chuan; Xia Lin-Hua; Wang Dong-Qing
2006-01-01
A state-observer based full-state asymptotic trajectory control (OFST3 method requiring a scalar state is presented to asymptotically drive all the states of chaotic systems to arbitrary desired trajectories. It is no surprise that OFSTC can obtain good tracking performance as desired due to using a state-observer. Significantly OFSTC requires only a scalar state of chaotic systems. A sinusoidal wave and two chaotic variables were taken as illustrative tracking trajectories to validate that using OFSTC can make all the states of a unified chaotic system track the desired trajectories with high tracking accuracy and in a finite time. It is noted that this is the first time that the state-observer of chaotic systems is designed on the basis of Kharitonov's Theorem.
Institute of Scientific and Technical Information of China (English)
Yu Fei; Wang Chun-Hua; Yin Jin-Wen; Xu Hao
2011-01-01
In this paper,we propose a novel four-dimensional autonomous chaotic system.Of particular interest is that this novel system can generate one-,two,three- and four-wing chaotic attractors with the variation of a single parameter,and the multi-wing type of the chaotic attractors can be displayed in all directions.The system is simple with a large positive Lyapunov exponent and can exhibit some interesting and complicated dynamical behaviours.Basic dynamical properties of the four-dimensional chaotic system,such as equilibrium points,the Poincaré map,the bifurcation diagram and the Lyapunov exponents are investigated by using either theoretical analysis or numerical method.Finally,a circuit is designed for the implementation of the multi-wing chaotic attractors.The electronic workbench observations are in good agreement with the numerical simulation results.
Energy Technology Data Exchange (ETDEWEB)
Xu Yuhua [College of Information Science and Technology, Donghua University, Shanghai 201620 (China) and Department of Maths, Yunyang Teacher' s College, Hubei 442000 (China)], E-mail: yuhuaxu2004@163.com; Zhou Wuneng [College of Information Science and Technology, Donghua University, Shanghai 201620 (China)], E-mail: wnzhou@163.com; Fang Jianan [College of Information Science and Technology, Donghua University, Shanghai 201620 (China)
2009-11-15
This paper introduces a modified Lue chaotic system, and some basic dynamical properties are studied. Based on these properties, we present hybrid dislocated control method for stabilizing chaos to unstable equilibrium and limit cycle. In addition, based on the Lyapunov stability theorem, general hybrid projective dislocated synchronization (GHPDS) is proposed, which includes complete dislocated synchronization, dislocated anti-synchronization and projective dislocated synchronization as its special item. The drive and response systems discussed in this paper can be strictly different dynamical systems (including different dimensional systems). As examples, the modified Lue chaotic system, Chen chaotic system and hyperchaotic Chen system are discussed. Numerical simulations are given to show the effectiveness of these methods.
Synchronization and anti-synchronization of chaotic systems: A differential and algebraic approach
Energy Technology Data Exchange (ETDEWEB)
Martinez-Guerra, Rafael [Departamento de Control Automatico, Cinvestav-IPN A. P. 14-740, Av. IPN 2508, 07360 Mexico, D.F. (Mexico)], E-mail: rguerra@ctrl.cinvestav.mx; Pasaye, Jose Juan Rincon [Departamento de Control Automatico, Cinvestav-IPN A. P. 14-740, Av. IPN 2508, 07360 Mexico, D.F. (Mexico)], E-mail: jrincon@ctrl.cinvestav.mx
2009-10-30
Chaotic systems synchronization and anti-synchronization problems are tackled by means of differential and algebraic techniques for nonlinear systems. An algebraic observer is proposed for systems satisfying an algebraic observability condition. This observer can be used as a slave system whose states are synchronized with the master (chaotic) system. This approach has the advantages of being independent of the chaotic nature of the master system, it uses a reduced set of measurable signal from the master system and it also solves the anti-synchronization problem as a straightforward extension of the synchronization one. A Colpitts oscillator is given to illustrate the effectiveness of the suggested approach.
Analysis and Anti-Synchronization of a Novel Chaotic System via Active and Adaptive Controllers
Directory of Open Access Journals (Sweden)
V. Sundarapandian
2013-09-01
Full Text Available Anti-synchronization of chaotic systems deals with the problem of asymptotically synchronizing the sum of states of a pair of chaotic systems called master and slave systems with the help of controllers attached to the slave system. When two chaotic systems are anti-synchronized, then their states are asymptotically equal in magnitude, but opposite in phase. Anti-synchronization of chaotic systems has applications in many engineering areas such as secure communications, secure data encryption, cryptosystems, etc. This paper announces a novel 3-D chaotic system and describes its qualitative properties. Next, this paper deals with the design of active and adaptive controllers for synchronizing the states of identical novel chaotic systems. Active controllers are used when the system parameters are available for measurement and the synchronization result is established using Lyapunov stability theory. Adaptive controllers are used when the system parameters are unknown. In this case, estimates are used in lieu of the unknown system parameters and adaptive controllers are designed using adaptive control theory and Lyapunov stability theory. Numerical simulations using MATLAB have been shown to demonstrate the proposed active and adaptive synchronization results for novel chaotic systems.
Institute of Scientific and Technical Information of China (English)
Qi Dong-Lian; Wang Qiao; Yang Jie
2011-01-01
Two different sliding mode controllers for a fractional order unified chaotic system are presented.The controller for an integer-order unified chaotic system is substituted directly into the fractional-order counterpart system,and the fractional-order system can be made asymptotically stable by this controller.By proving the existence of a sliding manifold containing fractional integral,the controller for a fractional-order system is obtained,which can stabilize it.A comparison between these different methods shows that the performance of a sliding mode controller with a fractional integral is more robust than the other for controlling a fractional order unified chaotic system.
A Novel Chaotic System without Equilibrium: Dynamics, Synchronization, and Circuit Realization
Directory of Open Access Journals (Sweden)
Ahmad Taher Azar
2017-01-01
Full Text Available A few special chaotic systems without unstable equilibrium points have been investigated recently. It is worth noting that these special systems are different from normal chaotic ones because the classical Shilnikov criterion cannot be used to prove chaos of such systems. A novel unusual chaotic system without equilibrium is proposed in this work. We discover dynamical properties as well as the synchronization of the new system. Furthermore, a physical realization of the system without equilibrium is also implemented to illustrate its feasibility.
Hybrid Projective Synchronization for Two Identical Fractional-Order Chaotic Systems
Directory of Open Access Journals (Sweden)
Ping Zhou
2012-01-01
Full Text Available A hybrid projective synchronization scheme for two identical fractional-order chaotic systems is proposed in this paper. Based on the stability theory of fractional-order systems, a controller for the synchronization of two identical fractional-order chaotic systems is designed. This synchronization scheme needs not to absorb all the nonlinear terms of response system. Hybrid projective synchronization for the fractional-order Chen chaotic system and hybrid projective synchronization for the fractional-order hyperchaotic Lu system are used to demonstrate the validity and feasibility of the proposed scheme.
Institute of Scientific and Technical Information of China (English)
LEI LiHua; SHI HuLi; MA GuanYi
2009-01-01
Multiple Path Interference (MPI) and Multiple Access Interference (MAI) are Important factors that affect the performance of Chinese Area Positioning System (CAPS),These problems can be solved by using spreading sequences with ideal properties and multi-user detectors.Chaotic sequences based on Chebyshev map are studied and the satellite communication system model is set up to investigate the application of chaotic sequences for CAPS in this paper,Simulation results show that chaotic sequences have desirable correlation properties and it is easy to generate a large number of chaotic sequences with good security.It has great practical value to apply chaotic sequences to CAPS together with multi-user detecting technology and the system performance can be improved greatly.
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
Multiple Path Interference (MPI) and Multiple Access Interference (MAI) are important factors that affect the performance of Chinese Area Positioning System (CAPS). These problems can be solved by using spreading sequences with ideal properties and multi-user detectors. Chaotic sequences based on Chebyshev map are studied and the satellite communication system model is set up to investigate the application of chaotic sequences for CAPS in this paper. Simulation results show that chaotic sequences have desirable correlation properties and it is easy to generate a large number of chaotic sequences with good security. It has great practical value to apply chaotic sequences to CAPS together with multi-user detecting technology and the system performance can be improved greatly.
System for Information Encryption Implementing Several Chaotic Orbits
Directory of Open Access Journals (Sweden)
Jiménez-Rodríguez Maricela
2015-07-01
Full Text Available This article proposes a symmetric encryption algorithm that takes, as input value, the original information of length L, that when encoded, generates the ciphertext of greater length LM. A chaotic discrete system (logistic map is implemented to generate 3 different orbits: the first is used for applying a diffusion technique in order to mix the original data, the second orbit is combined with the mixed information and increases the length of L to LM, and with the third orbit, the confusion technique is implemented. The encryption algorithm was applied to encode an image which is then totally recovered by the keys used to encrypt and his respective, decrypt algorithm. The algorithm can encode any information, just dividing into 8 bits, it can cover the requirements for high level security, it uses 7 keys to encrypt and provides good encryption speed
Spectral Statistics in Chaotic Systems with Two Identical Connected Cells
Dittrich, T; Koboldt, G; Dittrich, Thomas; Schanz, Holger; Koboldt, Gerd
1998-01-01
Chaotic systems that decompose into two cells connected only by a narrow channel exhibit characteristic deviations of their quantum spectral statistics from the canonical random-matrix ensembles. The equilibration between the cells introduces an additional classical time scale that is manifest also in the spectral form factor. If the two cells are related by a spatial symmetry, the spectrum shows doublets, reflected in the form factor as a positive peak around the Heisenberg time. We combine a semiclassical analysis with an independent random-matrix approach to the doublet splittings to obtain the form factor on all time (energy) scales. Its only free parameter is the characteristic time of exchange between the cells in units of the Heisenberg time.
Localization in chaotic systems with a single-channel opening.
Lippolis, Domenico; Ryu, Jung-Wan; Kim, Sang Wook
2015-07-01
We introduce a single-channel opening in a random Hamiltonian and a quantized chaotic map: localization on the opening occurs as a sensible deviation of the wave-function statistics from the predictions of random matrix theory, even in the semiclassical limit. Increasing the coupling to the open channel in the quantum model, we observe a similar picture to resonance trapping, made of a few fast-decaying states, whose left (right) eigenfunctions are entirely localized on the (preimage of the) opening, and plentiful long-lived states, whose probability density is instead suppressed at the opening. For the latter, we derive and test a linear relation between the wave-function intensities and the decay rates, similar to the Breit-Wigner law. We then analyze the statistics of the eigenfunctions of the corresponding (discretized) classical propagator, finding a similar behavior to the quantum system only in the weak-coupling regime.
Decoherence in chaotic and integrable systems: a random matrix approach
Gorin, T.; Seligman, T. H.
2003-03-01
We study the influence of chaos and order on entanglement and decoherence. In view of applications in quantum computing and teleportation which should be able to work with arbitrarily complicated states, we pay particular attention to the behavior of random states. While studies with coherent states indicate that chaos accelerates decoherence and entanglement, we find that there is practically no difference between the chaotic and the integrable case, as far as random states are concerned. In the present studies we use unitary time evolution of the total system, and partial traces to emulate decoherence. Random matrix models are a natural choice to describe the dynamics of random states. The invariant aspects of chaos and order are then reflected in the different spectral statistics. We develop random matrix models for the evolution of entanglement for a large variety of situations, discussing the strong coupling case in full detail.
Decoherence in chaotic and integrable systems: a random matrix approach
Energy Technology Data Exchange (ETDEWEB)
Gorin, T.; Seligman, T.H
2003-03-17
We study the influence of chaos and order on entanglement and decoherence. In view of applications in quantum computing and teleportation which should be able to work with arbitrarily complicated states, we pay particular attention to the behavior of random states. While studies with coherent states indicate that chaos accelerates decoherence and entanglement, we find that there is practically no difference between the chaotic and the integrable case, as far as random states are concerned. In the present studies we use unitary time evolution of the total system, and partial traces to emulate decoherence. Random matrix models are a natural choice to describe the dynamics of random states. The invariant aspects of chaos and order are then reflected in the different spectral statistics. We develop random matrix models for the evolution of entanglement for a large variety of situations, discussing the strong coupling case in full detail.
Study on phase synchronization of stochastic chaotic system
Institute of Scientific and Technical Information of China (English)
Yang Xiao-Li; Xu Wei
2008-01-01
This paper detects and characterizes the diverse roles played by bounded noise in chaotic phase synchronization (CPS) of weakly coupled nonlinear stochastic systems. Analysis of a paradigmatic model of two bidirectional coupled three-level food chains is carried out by various statistical measures such as Shannon entropy and mutual information. The results indicate that inside the synchronous regime, CPS is considerably reduced under the influence of bounded noise; near the onset of phase synchronization, temporal phase locking is diversely changed with the increase of noise, i.e., either weak or strong noise also degrades the degree of CPS, while intermediate noise enhances CPS remarkably, and an optimal noise intensity is detected that maximizes the enhancement.
Generalized Combination Complex Synchronization for Fractional-Order Chaotic Complex Systems
Directory of Open Access Journals (Sweden)
Cuimei Jiang
2015-07-01
Full Text Available Based on two fractional-order chaotic complex drive systems and one fractional-order chaotic complex response system with different dimensions, we propose generalized combination complex synchronization. In this new synchronization scheme, there are two complex scaling matrices that are non-square matrices. On the basis of the stability theory of fractional-order linear systems, we design a general controller via active control. Additionally, by virtue of two complex scaling matrices, generalized combination complex synchronization between fractional-order chaotic complex systems and real systems is investigated. Finally, three typical examples are given to demonstrate the effectiveness and feasibility of the schemes.
Analysis of Chaotic Waveforms for Application to Active Sonar Systems
1993-06-01
Noise-Reduced Signal. Figure 4-2. Noise Reduction Power Spectra : (a) Power Spectrum of Lorenz Waveform; (b) Gaussian Noise; (c) Signal Plus Noise; and (d...dimension, theoretic entropy and Lyapunov exponent, are also described for completeness even though they are not used in this study. 2.5.1 Correlation...For each lical center, a data covariance matrix is formed using the nearest neighbors. Singular value decomposition ( SVD ) is then applied to the matrix
Parametric Control on Fractional-Order Response for Lü Chaotic System
Moaddy, K
2013-04-10
This paper discusses the influence of the fractional order parameter on conventional chaotic systems. These fractional-order parameters increase the system degree of freedom allowing it to enter new domains and thus it can be used as a control for such dynamical systems. This paper investigates the behaviour of the equally-fractional-order Lü chaotic system when changing the fractional-order parameter and determines the fractional-order ranges for chaotic behaviour. Five different parameter values and six fractional-order cases are discussed through this paper. Unlike the conventional parameters, as the fractional-order increases the system response begins with stability, passing by chaotic behaviour then reaches periodic response. As the system parameter α increases, a shift in the fractional order is required to maintain chaotic response.Therefore, the range of chaotic response can be expanded or minimized by controlling the fractional-order parameter. The non-standard finite difference method is used to solve the fractional-order Lü chaotic system numerically to validate these responses.
Adaptive control and synchronization of a fractional-order chaotic system
Indian Academy of Sciences (India)
Chunlai Li; Yaonan Tong
2013-04-01
In this paper, the chaotic dynamics of a three-dimensional fractional-order chaotic system is investigated. The lowest order for exhibiting chaos in the fractional-order system is obtained. Adaptive schemes are proposed for control and synchronization of the fractional-order chaotic system based on the stability theory of fractional-order dynamic systems. The presented schemes, which contain only a single-state variable, are simple and flexible. Numerical simulations are used to demonstrate the feasibility of the presented methods.
Indian Academy of Sciences (India)
KARIMA RABAH; SAMIR LADACI; MOHAMED LASHAB
2017-09-01
In this paper, a new design of fractional-order sliding mode control scheme is proposed for the synchronization of a class of nonlinear fractional-order systems with chaotic behaviour. The considered design approach provides a set of fractional-order laws that guarantee asymptotic stability of fractional-order chaotic systems in the sense of the Lyapunov stability theorem. Two illustrative simulation examples on the fractional-order Genesio–Tesi chaotic systems and the fractional-order modified Jerk systems are provided. These examples show the effectiveness and robustness of this control solution.
Method to modify random matrix theory using short-time behavior in chaotic systems.
Smith, A Matthew; Kaplan, Lev
2009-09-01
We discuss a modification to random matrix theory (RMT) eigenstate statistics that systematically takes into account the nonuniversal short-time behavior of chaotic systems. The method avoids diagonalization of the Hamiltonian, instead requiring only knowledge of short-time dynamics for a chaotic system or ensemble of similar systems. Standard RMT and semiclassical predictions are recovered in the limits of zero Ehrenfest time and infinite Heisenberg time, respectively. As examples, we discuss wave-function autocorrelations and cross correlations and show how the approach leads to a significant improvement in the accuracy for simple chaotic systems where comparison can be made with brute-force diagonalization.
Delayed Feedback Control of Bao Chaotic System Based on Hopf Bifurcation Analysis
Directory of Open Access Journals (Sweden)
Farhad Khellat
2014-11-01
Full Text Available This paper is concerned with bifurcation and chaos control in a new chaotic system recently introduced by Bao et al [9]. First a condition that the system has a Hopf bifurcation is derived. Then by applying delayed feedback controller, the chaotic system is forced to have a stable periodic orbit extracting from chaotic attractor. This is done by making Hopf bifurcation value of the open loop and the closed loop systems identical. Also by suitable tuning of the controller parameters, unstable equilibrium points become stable. Numerical simulations verify the results.
Energy Technology Data Exchange (ETDEWEB)
Chen, H.-H. [Department of Mechanical Engineering, HsiuPing Institute of Technology, Taichung 412, Taiwan (China)], E-mail: richard@mail.hit.edu.tw
2009-04-15
Liu chaotic systems exhibit two- or four-scroll attractors and are observed in a variety of engineering phenomena, including rigid body motion, brushless DC motor system and so forth. This study applies the Lyapunov stability theorem to identify the sufficient conditions for the asymptotic stability of the equilibrium points of Liu chaotic systems. A linear balanced feedback gain control method is then employed to design a controller to achieve the global synchronization of two identical four-scroll Liu chaotic systems. The feasibility and effectiveness of the proposed chaos stability and synchronization schemes are verified via numerical simulations.
A new theorem to synchronization of unified chaotic systems via adaptive control
Institute of Scientific and Technical Information of China (English)
Lequan Min; Jianyi Jing
2003-01-01
Chaos synchronization has been applied in secure communication, chemical reaction, biological systems, and information processing. A new theorem to synchronization of unified chaotic systems via adaptive control is proposed. The consutructive theorem provides the design scheme for adaptive controller such that a respond system can synchronize with respect to an uncertain drive system. One example for discontinuous chaotic system is proposed to illustrate the effectiveness and feasibility.
On a new time-delayed feedback control of chaotic systems
Energy Technology Data Exchange (ETDEWEB)
Tian Lixin [Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu, 212013 (China)], E-mail: tianlx@ujs.edu.cn; Xu Jun; Sun Mei; Li Xiuming [Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu, 212013 (China)
2009-01-30
In this paper, using the idea of the successive dislocation feedback method, a new time-delayed feedback control method called the successive dislocation time-delayed feedback control (SDTDFC) is designed. Firstly, the idea of SDTDFC is introduced. Then some analytic sufficient conditions of the chaos control from the SDTDFC approach are derived for stabilization. Finally, some established results are further clarified via a case study of the Lorenz system with the numerical simulations.
Octonionic Lorenz-like condition
Indian Academy of Sciences (India)
Murat Tanişli; Bernard Jancewicz
2012-02-01
In this study, the octonion algebra and its general properties are deﬁned by the Cayley–Dickson’s multiplication rules for octonion units. The ﬁeld equations, potential equations and Maxwell equations for electromagnetism are investigated with the octonionic equations and these equations can be compared with their vectorial representations. The potential and wave equations for ﬁelds with sources are also provided. By using Maxwell equations, a Lorenz-like condition is newly suggested for electromagnetism. The existing equations including the photon mass provide the most acknowledged Lorenz condition for the magnetic monopole and the source.
Design and implementation of a novel multi-scroll chaotic system
Institute of Scientific and Technical Information of China (English)
Zhang Chao-Xia; Yu Si-Min
2009-01-01
This paper proposes a novel approach for generating a multi-scroll chaotic system. Together with the theoretical design and numerical simulations, three different types of attractor are available, governed by constructing triangular wave, sawtooth wave and hysteresis sequence. The presented new multi-scroll chaotic system is different from the classical multi-scroll chaotic Chua system in dimensionless state equations, nonlinear functions and maximum Lyapunov exponents. In addition, the basic dynamical behaviours, including equilibrium points, eigenvalues, eigenvectors, eigenplanes, bifurcation diagrams and Lyapunov exponents, are further investigated. The success of the design is illustrated by both numerical simulations and circuit experiments.
Institute of Scientific and Technical Information of China (English)
Zhang Ruo-Xun; Yang Shi-Ping
2012-01-01
In this paper we investigate the synchronization of a class of three-dimensional fractional-order chaotic systems.Based on the Lyapunov stability theory and adaptive control technique,a single adaptive-feedback controller is developed to synchronize a class of fractional-order chaotic systems.The presented controller which only contains a single driving variable is simple both in design and in implementation.Numerical simulation and circuit experimental results for fractional-order chaotic system are provided to illustrate the effectiveness of the proposed scheme.
Constraint on periodic orbits of chaotic systems given by Random Matrix Theory
Monastra, Alejandro G
2010-01-01
Considering the fluctuations of spectral functions, we prove that if chaotic systems fulfill the Bohigas-Gianonni-Schmit (BGS) conjecture, which relates their spectral statistics to that of random matrices, therefore by virtue of Gutzwiller trace formula, the instability of classical periodic orbits is constrained. In particular for two-dimensional chaotic systems, the Lyapunov exponent $\\lambda_p$ of each periodic orbit $p$ should be bigger than a minimum value $\\lambda_{\\text{min}} \\geq 0.850738$. This opens the possibility of new constraints for a system to be fully chaotic, or the failure of the BGS conjecture.
Institute of Scientific and Technical Information of China (English)
牟静; 陶超; 杜功焕
2003-01-01
In this paper we propose and investigate the synchronization of a new chaotic model with time-varying parameters and apply it to improve the security of chaotic communication. In this model, the chaotic system is modulated by both the message and the varying parameters. The varying parameters distort the phase space so heavily that they prevent the carrier from being broken by nonlinear dynamic forecasting method. Theory and simulation experiments with speech signal communication indicate that the receiver can gain a perfect synchronization with the transmitter, and the intruder cannot break down this communication system. We also discuss the robustness of the new communication system.
Institute of Scientific and Technical Information of China (English)
Mujing; TaoChao; DuGong-Huan
2003-01-01
In this paper we propose and investigate the synchronization of a new chaotic model with time-varying parameters and apply it to improve the security of chaotic communication. In this model, the chaotic system is modulated by both the message and the varying parameters. The varying parameters distort the phase space so heavily that they prevent the carrier from being broken by nonlinear dynamic forecasting method. Theory and simulation experiments with speech signal communication indicate that the receiver can gain a perfect synchronization with the transmitter, and the intruder cannot break down this communication system. We also discuss the robustness of the new communication system.
Directory of Open Access Journals (Sweden)
Neng-Sheng Pai
2010-01-01
Full Text Available This paper focuses on the chaos control problem of the unified chaotic systems with structured uncertainties. Applying Schur-complement and some matrix manipulation techniques, the controlled uncertain unified chaotic system is then transformed into the linear matrix inequality (LMI form. Based on Lyapunov stability theory and linear matrix inequality (LMI formulation, a simple linear feedback control law is obtained to enforce the prespecified exponential decay dynamics of the uncertain unified chaotic system. Numerical results validate the effectiveness of the proposed robust control scheme.
Sundarapandian Vaidyanathan
2013-01-01
In this paper, we apply backstepping control method to derive new results for the adaptive controller andsynchronizer design for the Arneodo chaotic system (1980), when the system parameters are unknown.First, we design an adaptive backstepping controller to stabilize the Arneodo system to its unstableequilibrium at the origin. Next, we design an adaptive backstepping controller to achieve global chaossynchronization of the identical Arneodo chaotic systems with unknown parameters. MATLAB sim...
Estimating model parameters in nonautonomous chaotic systems using synchronization
Yang, Xiaoli; Xu, Wei; Sun, Zhongkui
2007-05-01
In this Letter, a technique is addressed for estimating unknown model parameters of multivariate, in particular, nonautonomous chaotic systems from time series of state variables. This technique uses an adaptive strategy for tracking unknown parameters in addition to a linear feedback coupling for synchronizing systems, and then some general conditions, by means of the periodic version of the LaSalle invariance principle for differential equations, are analytically derived to ensure precise evaluation of unknown parameters and identical synchronization between the concerned experimental system and its corresponding receiver one. Exemplifies are presented by employing a parametrically excited 4D new oscillator and an additionally excited Ueda oscillator. The results of computer simulations reveal that the technique not only can quickly track the desired parameter values but also can rapidly respond to changes in operating parameters. In addition, the technique can be favorably robust against the effect of noise when the experimental system is corrupted by bounded disturbance and the normalized absolute error of parameter estimation grows almost linearly with the cutoff value of noise strength in simulation.
Estimating model parameters in nonautonomous chaotic systems using synchronization
Energy Technology Data Exchange (ETDEWEB)
Yang, Xiaoli [Department of Applied Mathematics, Northwestern Polytechnical University, Xi' an 710072 (China)]. E-mail: yangxl205@mail.nwpu.edu.cn; Xu, Wei [Department of Applied Mathematics, Northwestern Polytechnical University, Xi' an 710072 (China); Sun, Zhongkui [Department of Applied Mathematics, Northwestern Polytechnical University, Xi' an 710072 (China)
2007-05-07
In this Letter, a technique is addressed for estimating unknown model parameters of multivariate, in particular, nonautonomous chaotic systems from time series of state variables. This technique uses an adaptive strategy for tracking unknown parameters in addition to a linear feedback coupling for synchronizing systems, and then some general conditions, by means of the periodic version of the LaSalle invariance principle for differential equations, are analytically derived to ensure precise evaluation of unknown parameters and identical synchronization between the concerned experimental system and its corresponding receiver one. Exemplifies are presented by employing a parametrically excited 4D new oscillator and an additionally excited Ueda oscillator. The results of computer simulations reveal that the technique not only can quickly track the desired parameter values but also can rapidly respond to changes in operating parameters. In addition, the technique can be favorably robust against the effect of noise when the experimental system is corrupted by bounded disturbance and the normalized absolute error of parameter estimation grows almost linearly with the cutoff value of noise strength in simulation.
Chaotic signal reconstruction with application to noise radar system
Liu, Lidong; Hu, Jinfeng; He, Zishu; Han, Chunlin; Li, Huiyong; Li, Jun
2011-12-01
Chaotic signals are potentially attractive in engineering applications, most of which require an accurate estimation of the actual chaotic signal from a noisy background. In this article, we present an improved symbolic dynamics-based method (ISDM) for accurate estimating the initial condition of chaotic signal corrupted by noise. Then, a new method, called piecewise estimation method (PEM), for chaotic signal reconstruction based on ISDM is proposed. The reconstruction performance using PEM is much better than that using the existing initial condition estimation methods. Next, PEM is applied in a noncoherent reception noise radar scheme and an improved noncoherent reception scheme is given. The simulation results show that the improved noncoherent scheme has better correlation performance and range resolution especially at low signal-to-noise ratios (SNRs).
Chaotic signal reconstruction with application to noise radar system
Directory of Open Access Journals (Sweden)
Liu Lidong
2011-01-01
Full Text Available Abstract Chaotic signals are potentially attractive in engineering applications, most of which require an accurate estimation of the actual chaotic signal from a noisy background. In this article, we present an improved symbolic dynamics-based method (ISDM for accurate estimating the initial condition of chaotic signal corrupted by noise. Then, a new method, called piecewise estimation method (PEM, for chaotic signal reconstruction based on ISDM is proposed. The reconstruction performance using PEM is much better than that using the existing initial condition estimation methods. Next, PEM is applied in a noncoherent reception noise radar scheme and an improved noncoherent reception scheme is given. The simulation results show that the improved noncoherent scheme has better correlation performance and range resolution especially at low signal-to-noise ratios (SNRs.
Hyper-chaotic LÜ System Simulation Design of Digital Circuit Based on DSP Builder
Directory of Open Access Journals (Sweden)
Zhang Xiaohong
2013-05-01
Full Text Available In order to overcome sensitive defects of components deviations and environment defects in analog circuit design, a novel four-dimensional hyperchaotic systems is constructed based on LÜ system. Basic nonlinear dynamics characteristics are analyzed to the new system. By optimizing the design of sampling frequency selection, gain adjustments, parameter configuration etc., the hyperchaotic circuit runs stably while the signal amplitude is reasonably controlled. Curves of the digital chaotic sequence show smooth without jagged shape. The circuit can be applied to other digital realization of chaotic systems with commonality and expandability. Its experimental results fully consistent with phase space structures of the continuous chaotic system, which shows the development of chaotic systems based on FPGA is feasible and practical.
Fuzzy Modeling, Tracking Control and Synchronization of the Rossler's Chaotic System
Institute of Scientific and Technical Information of China (English)
方建安; 范丹丹
2004-01-01
In this paper, a novel method to model, track control and synchronize the Rossler's chaotic system is proposed. The fuzzy logical system is used so that the fuzzy inference rule is transferred into a type of variable coef ficient nonlinear ordinary differential equation. Consequently the model of the chaotic system is obtained. Then a fuzzy tracking control and a fuzzy synchronization for chaotic systems is proposed as well. First, a known tracking control for the Rossler's system is used in this paper. We represent the Rossler's chaotic and control systems into fuzzy inference rules. Then the variable coefficient nonlinear ordinary differential equation is also got. Simulation results show that such an approach is effective and has a high precision.
Generalized Projective Synchronization of Fractional Order Chaotic Systems with Different Dimensions
Institute of Scientific and Technical Information of China (English)
WANG Sha; YU Yong-Guang
2012-01-01
The generalized projective synchronization of different dimensional fractional order chaotic systems is investigated. According to the stability theory of linear fractional order systems, a sufficient condition to realize synchronization is obtained. The fractional order chaotic and hyperchaotic systems are applied to achieve synchronization in both reduced and increased dimensions. The corresponding numerical results coincide with theoretical analysis.%The generalized projective synchronization of different dimensional fractional order chaotic systems is investigated.According to the stability theory of linear fractional order systems,a sufficient condition to realize synchronization is obtained.The fractional order chaotic and hyperchaotic systems are applied to achieve synchronization in both reduced and increased dimensions.The corresponding numerical results coincide with theoretical analysis.
Synchronisation of chaotic systems using a novel sampled-data fuzzy controller
Institute of Scientific and Technical Information of China (English)
Feng Yi-Fu; Zhang Qing-Ling
2011-01-01
This paper presents the synchronisation of chaotic systems using a sampled-data fuzzy controller and is meaningful for many physical real-life applications. Firstly, a Takagi-Sugeno (T-S) fuzzy model is employed to represent the chaotic systems that contain some nonlinear terms, then a type of fuzzy sampled-data controller is proposed and an error system formed by the response and drive chaotic system. Secondly, relaxed LMI-based synchronisation conditions are derived by using a new paraneter-dependent Lyapunov-Krasovskii functional and relaxed stabilisation techniques for the underlying error system. The derived LMI-based conditions are used to aid the design of a sampled-data fuzzy controller to achieve the synchronisation of chaotic systems. Finally, a numerical example is provided to illustrate the effectiveness of the proposed results.
A description of stochastic systems using chaotic maps
Directory of Open Access Journals (Sweden)
Abraham Boyarsky
2004-01-01
Full Text Available Let ρ(x,t denote a family of probability density functions parameterized by time t. We show the existence of a family {τ1:t>0} of deterministic nonlinear (chaotic point transformations whose invariant probability density functions are precisely ρ(x,t. In particular, we are interested in the densities that arise from the diffusions. We derive a partial differential equation whose solution yields the family of chaotic maps whose density functions are precisely those of the diffusion.
Scar and Antiscar Quantum Effects in Open Chaotic Systems
Kaplan, L
1999-01-01
We predict and numerically observe strong periodic orbit effects in the properties of open quantum systems with a chaotic classical limit. Antiscars lead to a large number of exponentially narrow resonances when the opening is located on a short unstable orbit of the closed system; the probability to remain in the system at long times is thus exponentially enhanced over the random matrix theory prediction. The distribution of resonance widths and the probability to remain are quantitatively given in terms of only the stability matrix of the orbit on which the opening is placed. The long-time remaining probability density is non-trivially distributed over the available phase space; it can be enhanced or suppressed near orbits other than the one on which the lead is located, depending on the periods and classical actions of these other orbits. These effects of the short periodic orbits on quantum decay rates have no classical counterpart, and first appear on times scales much larger than the Heisenberg time of ...
Anticipating synchronization in a chain of chaotic oscillators with switching parameters
Energy Technology Data Exchange (ETDEWEB)
Pyragienė, T., E-mail: tatjana.pyragiene@ftmc.lt; Pyragas, K.
2015-12-18
A new coupling scheme for anticipating synchronization of chaotic systems is proposed. The scheme consists of a master system and two in series coupled slave systems with periodically switching parameters. The scheme does not require the presence of any time-delay terms either in a master or in slave systems and provides long-term anticipation. The value of anticipation time as well as the conditions of synchronization are derived in an analytical form. Analytical results are tested by numerical experiments with the chaotic Rössler and Lorenz systems as well as the Hindmarsh–Rose neuron in a regime of chaotic bursting. Also a robustness of the scheme with respect to parameter mismatch and noise is demonstrated. - Highlights: • A new coupling scheme for anticipating chaotic synchronization is proposed. • The scheme consists of three coupled systems with periodically switching parameters. • Long-term anticipation is achieved without using time-delay terms. • The method is verified for the Rössler, Lorenz and Hindmarsh–Rose neuron systems.
Hopf Bifurcation Analysis and Chaos Control of a Chaotic System without ilnikov Orbits
Directory of Open Access Journals (Sweden)
Na Li
2015-01-01
Full Text Available This paper mainly investigates the dynamical behaviors of a chaotic system without ilnikov orbits by the normal form theory. Both the stability of the equilibria and the existence of local Hopf bifurcation are proved in view of analyzing the associated characteristic equation. Meanwhile, the direction and the period of bifurcating periodic solutions are determined. Regarding the delay as a parameter, we discuss the effect of time delay on the dynamics of chaotic system with delayed feedback control. Finally, numerical simulations indicate that chaotic oscillation is converted into a steady state when the delay passes through a certain critical value.
Further results on complete synchronization for noise-perturbed chaotic systems
Zhou, Jie; Chen, Zhang
2008-08-01
In this Letter, a class of general systems which covers several famous chaotic systems is studied in complete synchronization with noise perturbation. Special nonlinear coupling techniques as well as LaSalle-type invariance principle of stochastic differential equations are employed to deduce our sufficient conditions for complete synchronization without involving the boundedness of chaotic systems. Furthermore, the correlative numerical simulations are given to illustrate the effectiveness of our theoretic results.
Further results on complete synchronization for noise-perturbed chaotic systems
Energy Technology Data Exchange (ETDEWEB)
Zhou Jie [Research Center and Laboratory of Mathematics for Nonlinear Science, School of Mathematical Sciences, Fudan University, Shanghai 200433 (China)], E-mail: 031018020@fudan.edu.cn; Chen Zhang [School of Mathematics, Shandong University, Jinan 250100 (China)], E-mail: chenzhangcz@163.com
2008-08-11
In this Letter, a class of general systems which covers several famous chaotic systems is studied in complete synchronization with noise perturbation. Special nonlinear coupling techniques as well as LaSalle-type invariance principle of stochastic differential equations are employed to deduce our sufficient conditions for complete synchronization without involving the boundedness of chaotic systems. Furthermore, the correlative numerical simulations are given to illustrate the effectiveness of our theoretic results.
Chaotic anti-control for the bounded linear continuous-time system
Institute of Scientific and Technical Information of China (English)
无
2008-01-01
With regard to the bounded linear continuous-time system,a universal chaotic anti-controlling method was presented on the basis of tracking control.A tracking controller is designed to such an extent that it can track any chaotic reference input,thus making it possible to chaotify the linear system.The controller is identical in structure for different controlled linear systems.Computer simulations proved the effectiveness of the proposed method.
Chaotic anti-control for the bounded linear continuous-time system
Institute of Scientific and Technical Information of China (English)
Li Jianfen; Lin Hui; Li Nong
2008-01-01
With regard to the bounded linear continuous-time system, a universal chaotic anti-controlling method was presented on the basis of tracking control. A tracking controller is designed to such an extent that it can track any chaotic reference input, thus making it possible to chaotify the linear system. The controller is identical in structure for different controlled linear systems. Computer simulations proved the effectiveness of the proposed method.
Increasing-order Projective Synchronization of Chaotic Systems with Time Delay
Institute of Scientific and Technical Information of China (English)
MIAO Qing-Ying; FANG Jian-An; TANG Yang; DONG Ai-Hua
2009-01-01
This work is concerned with lag projective synchronization of chaotic systems with increasing order. The systems under consideration have unknown parameters and different structures. Combining the adaptive control method and feedback control technique, we design a suitable controller and parameter update law to achieve lag synchronization of chaotic systems with increasing order. The result is rigorously proved by the Lyapunov stability theorem. Moreover, corresponding simulation results are given to verify the effectiveness of the proposed methods.
Sliding Mode Control of the Fractional-Order Unified Chaotic System
Jian Yuan; Bao Shi; Xiaoyun Zeng; Wenqiang Ji; Tetie Pan
2013-01-01
This paper deals with robust synchronization of the fractional-order unified chaotic systems. Firstly, control design for synchronization of nominal systems is proposed via fractional sliding mode technique. Then, systematic uncertainties and external disturbances are considered in the fractional-order unified chaotic systems, and adaptive sliding mode control is designed for the synchronization issue. Finally, numerical simulations are carried out to verify the effectiveness of the two propo...
Generalized Combination Complex Synchronization for Fractional-Order Chaotic Complex Systems
Cuimei Jiang; Shutang Liu; Da Wang
2015-01-01
Based on two fractional-order chaotic complex drive systems and one fractional-order chaotic complex response system with different dimensions, we propose generalized combination complex synchronization. In this new synchronization scheme, there are two complex scaling matrices that are non-square matrices. On the basis of the stability theory of fractional-order linear systems, we design a general controller via active control. Additionally, by virtue of two complex scaling matrices, general...
Chaotic system for the detection of periodic signals under the background of strong noise
Institute of Scientific and Technical Information of China (English)
无
2003-01-01
We propose a method to study the chaotic system for the detection of periodic signals in the presence of strong background noise. The numerical experiments indicate that the chaotic system constructed from the modified Duffing-Holmes equation is sensitive to the weak periodic signal mixed with noise, and it has certain immunity to noise. The signal to noise ratio for the system can reach to about -91 dB.
Energy Technology Data Exchange (ETDEWEB)
Santos Coelho, Leandro dos [Pontifical Catholic University of Parana, PUCPR Industrial and Systems Engineering Graduate Program, LAS/PPGEPS Imaculada Conceicao, 1155, Zip code 80215-901, Curitiba, Parana (Brazil)], E-mail: leandro.coelho@pucpr.br
2009-02-28
Despite the popularity, the tuning aspect of proportional-integral-derivative (PID) controllers is a challenge for researchers and plant operators. Various controllers tuning methodologies have been proposed in the literature such as auto-tuning, self-tuning, pattern recognition, artificial intelligence, and optimization methods. Chaotic optimization algorithms as an emergent method of global optimization have attracted much attention in engineering applications. Chaotic optimization algorithms, which have the features of easy implementation, short execution time and robust mechanisms of escaping from local optimum, is a promising tool for engineering applications. In this paper, a tuning method for determining the parameters of PID control for an automatic regulator voltage (AVR) system using a chaotic optimization approach based on Lozi map is proposed. Since chaotic mapping enjoys certainty, ergodicity and the stochastic property, the proposed chaotic optimization introduces chaos mapping using Lozi map chaotic sequences which increases its convergence rate and resulting precision. Simulation results are promising and show the effectiveness of the proposed approach. Numerical simulations based on proposed PID control of an AVR system for nominal system parameters and step reference voltage input demonstrate the good performance of chaotic optimization.
LMI-based output feedback fuzzy control of chaotic system with uncertainties
Institute of Scientific and Technical Information of China (English)
Tan Wen; Wang Yao-Nan; Duan Feng; Li Xiao-Hui
2006-01-01
This paper studies the robust fuzzy control for nonlinear chaotic system in the presence of parametric uncertainties. An uncertain Takagi-Sugeno (T-S) fuzzy model is employed for fuzzy modelling of an unknown chaotic system. A sufficient condition formulated in terms of linear matrix inequality (LMI) for the existence of fuzzy controller is obtained. Then the output feedback fuzzy-model-based regulator derived from the LMI solutions can guarantee the stability of the closed-loop overall fuzzy system. The T-S fuzzy model of the chaotic Chen system is developed as an example for illustration. The effectiveness of the proposed controller design methodology is finally demonstrated through computer simulations on the uncertain Chen chaotic system.
Institute of Scientific and Technical Information of China (English)
王贺元
2012-01-01
To discuss the dynamical behavior of the flow between two concentric rotating spheres we study the dynamical behavior and the numerical simulation of the model system similar to the Lorenz equations of the Navier-Stokes equations for the flow between two concentric rotating spheres. Its stationary points and the stability are presented, the existence of attractor is proved, and the global stability of the system is discussed. Chaos behavior is simulated numerically by computer with the changing of Reynolds number.
A theory of nonequilibrium steady states in quantum chaotic systems
Wang, Pei
2017-09-01
Nonequilibrium steady state (NESS) is a quasistationary state, in which exist currents that continuously produce entropy, but the local observables are stationary everywhere. We propose a theory of NESS under the framework of quantum chaos. In an isolated quantum system whose density matrix follows a unitary evolution, there exist initial states for which the thermodynamic limit and the long-time limit are noncommutative. The density matrix \\hat ρ of these states displays a universal structure. Suppose that \\renewcommand{\\ket}[1]{{\\vert #1 >}} \\ketα and \\renewcommand{\\ket}[1]{{\\vert #1 >}} \\ketβ are different eigenstates of the Hamiltonian with energies E_α and E_β , respectively. \\renewcommand{\\bra}[1]{} \\braα\\hat ρ \\ketβ behaves as a random number which has zero mean. In thermodynamic limit, the variance of \\renewcommand{\\bra}[1]{} \\braα\\hat ρ \\ketβ is a smooth function of ≤ft\\vert E_α-E_β\\right\\vert , scaling as 1/≤ft\\vert E_α-E_β\\right\\vert 2 in the limit ≤ft\\vert E_α-E_β\\right\\vert \\to 0 . If and only if this scaling law is obeyed, the initial state evolves into NESS in the long time limit. We present numerical evidence of our hypothesis in a few chaotic models. Furthermore, we find that our hypothesis indicates the eigenstate thermalization hypothesis (ETH) for current operators in a bipartite system.
Wen-Bo, Wang; Xiao-Dong, Zhang; Yuchan, Chang; Xiang-Li, Wang; Zhao, Wang; Xi, Chen; Lei, Zheng
2016-01-01
In this paper, a new method to reduce noises within chaotic signals based on ICA (independent component analysis) and EMD (empirical mode decomposition) is proposed. The basic idea is decomposing chaotic signals and constructing multidimensional input vectors, firstly, on the base of EMD and its translation invariance. Secondly, it makes the independent component analysis on the input vectors, which means that a self adapting denoising is carried out for the intrinsic mode functions (IMFs) of chaotic signals. Finally, all IMFs compose the new denoised chaotic signal. Experiments on the Lorenz chaotic signal composed of different Gaussian noises and the monthly observed chaotic sequence on sunspots were put into practice. The results proved that the method proposed in this paper is effective in denoising of chaotic signals. Moreover, it can correct the center point in the phase space effectively, which makes it approach the real track of the chaotic attractor. Project supported by the National Science and Technology, China (Grant No. 2012BAJ15B04), the National Natural Science Foundation of China (Grant Nos. 41071270 and 61473213), the Natural Science Foundation of Hubei Province, China (Grant No. 2015CFB424), the State Key Laboratory Foundation of Satellite Ocean Environment Dynamics, China (Grant No. SOED1405), the Hubei Provincial Key Laboratory Foundation of Metallurgical Industry Process System Science, China (Grant No. Z201303), and the Hubei Key Laboratory Foundation of Transportation Internet of Things, Wuhan University of Technology, China (Grant No.2015III015-B02).
基于高斯伪谱方法的混沌系统最优控制%Optimal control for a chaotic system by means of Gauss pseudospectral method
Institute of Scientific and Technical Information of China (English)
曹小群
2013-01-01
针对混沌系统最优控制问题，提出一种基于高斯伪谱方法的数值求解新算法。首先在勒让德-高斯点上构造Lagrange插值多项式并近似表示混沌系统最优控制中的状态变量和控制变量；接着将连续空间的最优控制问题转化为非线性规划问题；然后通过序列二次规划(SQP)算法获得最优解；最后对三个典型混沌系统的仿真实验结果表明，新方法能有效和快速地实现混沌系统的最优控制。%A new numerical method is presented to solve optimal control problem of a chaotic system based on Gauss pseudospectral method (GPM). Firstly, the Lagrange interpolation polynomials are constructed on Legendre-Gauss nodes and used to parameterize the state and control the trajectories in optimal control of the chaotic system. Then, the chaotic optimal control problem in the continuous space is transformed into a nonlinear programming (NLP) problem through GPM. Furthermore, the NLP problem is solved by the sequential quadratic programming algorithm. Finally, the proposed method is applied to the optimal control of the typical Lorenz, Chen, and Liu chaotic systems respectively. The simulation processes indicate that the GPM is effective, fast and feasible for solving optimal control problems of chaotic systems.
The open-plus-closed loop (OPCL) method for chaotic systems with multiple strange attractors
Institute of Scientific and Technical Information of China (English)
Song Yun-Zhong
2007-01-01
Based on the open-plus-closed-loop (OPCL) control method a systematic and comprehensive controller is presented in this paper for a chaotic system, that is, the Newton-Leipnik equation with two strange attractors: the upper attractor(UA) and the lower attractor (LA). Results show that the final structure of the suggested controller for stabilization has a simple linear feedback form. To keep the integrity of the suggested approach, the globality proof of the basins of entrainment is also provided. In virtue of the OPCL technique, three different kinds of chaotic controls of the system are investigated, separately: the original control forcing the chaotic motion to settle down to the origin from an arbitrary position of the phase space; the chaotic intra-attractor control for stabilizing the equilibrium points only belonging to the upper chaotic attractor or the lower chaotic one; and the inter-attractor control for compelling the chaotic oscillation from one basin to another one. Both theoretical analysis and simulation results verify the validity of the proposed means.
NOISE-INDUCED CHAOTIC MOTIONS IN HAMILTONIAN SYSTEMS WITH SLOW-VARYING PARAMETERS
Institute of Scientific and Technical Information of China (English)
王双连; 郭乙木; 甘春标
2001-01-01
This paper studies chaotic motions in quasi-integrable Hamiltonian systems with slow-varying parameters under both harmonic and noise excitations.Based on the dynamic theory and some assumptions of excited noises, an extended form of the stochastic Melnikov method is presented. Using this extended method, the homoclinic bifurcations and chaotic behavior of a nonlinear Hamiltonian system with weak feed-back control under both harmonic and Gaussian white noise excitations are analyzed in detail. It is shown that the addition of stochastic excitations can make the parameter threshold value for the occurrence of chaotic motions vary in a wider region. Therefore, chaotic motions may arise easily in the system. By the Monte-Carlo method, the numerical results for the time-history and the maximum Lyapunov exponents of an example system are finally given to illustrate that the presented method is effective.
Runge-Kutta model-based nonlinear observer for synchronization and control of chaotic systems.
Beyhan, Selami
2013-07-01
This paper proposes a novel nonlinear gradient-based observer for synchronization and observer-based control of chaotic systems. The model is based on a Runge-Kutta model of the chaotic system where the evolution of the states or parameters is derived based on the error-square minimization. The stability and convergence conditions of observer and control methods are analyzed using a Lyapunov stability approach. In numerical simulations, the proposed observer and well-known sliding-mode observer are compared for the synchronization of a Lü chaotic system and observer-based stabilization of a Chen chaotic system. The noisy case for synchronization and parameter uncertainty case for stabilization are also considered for both observer-based methods. Copyright © 2013 ISA. Published by Elsevier Ltd. All rights reserved.
Attractors of relaxation discrete-time systems with chaotic dynamics on a fast time scale
Energy Technology Data Exchange (ETDEWEB)
Maslennikov, Oleg V.; Nekorkin, Vladimir I. [Institute of Applied Physics of RAS, Nizhny Novgorod (Russian Federation)
2016-07-15
In this work, a new type of relaxation systems is considered. Their prominent feature is that they comprise two distinct epochs, one is slow regular motion and another is fast chaotic motion. Unlike traditionally studied slow-fast systems that have smooth manifolds of slow motions in the phase space and fast trajectories between them, in this new type one observes, apart the same geometric objects, areas of transient chaos. Alternating periods of slow regular motions and fast chaotic ones as well as transitions between them result in a specific chaotic attractor with chaos on a fast time scale. We formulate basic properties of such attractors in the framework of discrete-time systems and consider several examples. Finally, we provide an important application of such systems, the neuronal electrical activity in the form of chaotic spike-burst oscillations.
Application of Multistage Homotopy Perturbation Method to the Chaotic Genesio System
Directory of Open Access Journals (Sweden)
M. S. H. Chowdhury
2012-01-01
Full Text Available Finding accurate solution of chaotic system by using efficient existing numerical methods is very hard for its complex dynamical behaviors. In this paper, the multistage homotopy-perturbation method (MHPM is applied to the Chaotic Genesio system. The MHPM is a simple reliable modification based on an adaptation of the standard homotopy-perturbation method (HPM. The HPM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions to the Chaotic Genesio system. Numerical comparisons between the MHPM and the classical fourth-order Runge-Kutta (RK4 solutions are made. The results reveal that the new technique is a promising tool for the nonlinear chaotic systems of ordinary differential equations.
Attractors of relaxation discrete-time systems with chaotic dynamics on a fast time scale.
Maslennikov, Oleg V; Nekorkin, Vladimir I
2016-07-01
In this work, a new type of relaxation systems is considered. Their prominent feature is that they comprise two distinct epochs, one is slow regular motion and another is fast chaotic motion. Unlike traditionally studied slow-fast systems that have smooth manifolds of slow motions in the phase space and fast trajectories between them, in this new type one observes, apart the same geometric objects, areas of transient chaos. Alternating periods of slow regular motions and fast chaotic ones as well as transitions between them result in a specific chaotic attractor with chaos on a fast time scale. We formulate basic properties of such attractors in the framework of discrete-time systems and consider several examples. Finally, we provide an important application of such systems, the neuronal electrical activity in the form of chaotic spike-burst oscillations.
Chaotic System Identification Based on a Fuzzy Wiener Model with Particle Swarm Optimization
Institute of Scientific and Technical Information of China (English)
LI Yong; TANG Ying-Gan
2010-01-01
@@ A fuzzy Wiener model is proposed to identify chaotic systems.The proposed fuzzy Wiener model consists of two parts,one is a linear dynamic subsystem and the other is a static nonlinear part,which is represented by the Takagi-Sugeno fuzzy model Identification of chaotic systems is converted to find optimal parameters of the fuzzy Wiener model by minimizing the state error between the original chaotic system and the fuzzy Wiener model.Particle swarm optimization algorithm,a global optimizer,is used to search the optimal parameter of the fuzzy Wiener model.The proposed method can identify the parameters of the linear part and nonlinear part simultaneously.Numerical simulations for Henón and Lozi chaotic system identification show the effectiveness of the proposed method.
Uniform semiclassical wave packet propagation and eigenstate extraction in a smooth chaotic system
Provost, D
1994-01-01
A uniform semiclassical propagator is used to time evolve a wavepacket in a smooth Hamiltonian system at energies for which the underlying classical motion is chaotic. The propagated wavepacket is Fourier transformed to yield a scarred eigenstate.
Dynamic Reconstruction-Based Fuzzy Neural Network Method for Fault Detection in Chaotic System
Institute of Scientific and Technical Information of China (English)
YANG Hongying; YE Hao; WANG Guizeng
2008-01-01
This paper presents a method for detecting weak fault signals in chaotic systems based on the chaotic dynamics reconstruction technique and the fuzzy neural system (FNS). The Grassberger-Procaccia algorithm and least squares regression were used to calculate the correlation dimension for the model order estimate. Based on the model order, an appropriately structured FNS model was designed to predict system faults. Through reasonable analysis of predicted errors, the disturbed signal can be extracted efficiently and correctly from the chaotic background. Satisfactory results were obtained by using several kinds of simula-tive faults which were extracted from the practical chaotic fault systems. Experimental results demonstra tethat the proposed approach has good prediction accuracy and can deal with data having a -40 dB signal to noise ratio (SNR). The low SNR requirement makes the approach a powerful tool for early fault detection.
Chaotic behaviour of nonlinear coupled reaction–diffusion system in four-dimensional space
Indian Academy of Sciences (India)
Li Zhang; Shutang Liu; Chenglong Yu
2014-06-01
In recent years, nonlinear coupled reaction–diffusion (CRD) system has been widely investigated by coupled map lattice method. Previously, nonlinear behaviour was observed dynamically when one or two of the three variables in the discrete system change. In this paper, we consider the chaotic behaviour when three variables change, which is called as four-dimensional chaos. When two parameters in the discrete system are unknown, we first give the existing condition of the chaos in four-dimensional space by the generalized definitions of spatial periodic orbits and spatial chaos. In addition, the chaotic behaviour will vary with the parameters. Then we propose a generalized Lyapunov exponent in four-dimensional space to characterize the different effects of parameters on the chaotic behaviour, which has not been studied in detail. In order to verify the chaotic behaviour of the system and the different effects clearly, we simulate the dynamical behaviour in two- and three-dimensional spaces.
Qiu, Junchao; Zhang, Lin; Li, Diyang; Liu, Xingcheng
2016-06-01
Chaotic sequences can be applied to realize multiple user access and improve the system security for a visible light communication (VLC) system. However, since the map patterns of chaotic sequences are usually well known, eavesdroppers can possibly derive the key parameters of chaotic sequences and subsequently retrieve the information. We design an advanced encryption standard (AES) interleaving aided multiple user access scheme to enhance the security of a chaotic code division multiple access-based visible light communication (C-CDMA-VLC) system. We propose to spread the information with chaotic sequences, and then the spread information is interleaved by an AES algorithm and transmitted over VLC channels. Since the computation complexity of performing inverse operations to deinterleave the information is high, the eavesdroppers in a high speed VLC system cannot retrieve the information in real time; thus, the system security will be enhanced. Moreover, we build a mathematical model for the AES-aided VLC system and derive the theoretical information leakage to analyze the system security. The simulations are performed over VLC channels, and the results demonstrate the effectiveness and high security of our presented AES interleaving aided chaotic CDMA-VLC system.
Chaos Control and Synchronization of a Novel Chaotic System Based upon Adaptive Control Algorithm
Directory of Open Access Journals (Sweden)
Israr Ahmad
2014-08-01
Full Text Available Controlling chaos is stabilizing one of the unstable periodic orbits either to its equilibrium point or to a stable periodic orbit by means of an appropriate continuous signal injected to the system. On the other hand, chaos synchronization refers to a procedure where two chaotic oscillators (either identical or nonidentical adjust a given property of their motion to a common behavior. This research paper concerns itself with the Adaptive Control and Synchronization of a new chaotic system with unknown parameters. Based on the Lyapunov Direct Method, the Adaptive Control Techniques are designed in such a way that the trajectory of the new chaotic system is globally stabilized to one of its equilibrium points of the uncontrolled system. Moreover, the Adaptive Control Law is also applied to achieve the synchronization state of two identical systems and two different chaotic systems with fully unknown parameters. The parameters identification, chaos control and synchronization of the chaotic system have been carried out simultaneously by the Adaptive Controller. All simulation results are carried out to corroborate the effectiveness and the robustness of the proposed methodology and possible feasibility for synchronizing two chaotic systems by using mathematica 9.
Sunada, Satoshi; Harayama, Takahisa; Davis, Peter; Tsuzuki, Ken; Arai, Ken-Ichi; Yoshimura, Kazuyuki; Uchida, Atsushi
2012-12-01
We present an experimental method for directly observing the amplification of microscopic intrinsic noise in a high-dimensional chaotic laser system, a laser with delayed feedback. In the experiment, the chaotic laser system is repeatedly switched from a stable lasing state to a chaotic state, and the time evolution of an ensemble of chaotic states starting from the same initial state is measured. It is experimentally demonstrated that intrinsic noises amplified by the chaotic dynamics are transformed into macroscopic fluctuating signals, and the probability density of the output light intensity actually converges to a natural invariant probability density in a strongly chaotic regime. Moreover, with the experimental method, we discuss the application of the chaotic laser systems to physical random bit generators. It is experimentally shown that the convergence to the invariant density plays an important role in nondeterministic random bit generation, which could be desirable for future ultimate secure communication systems.
Neng-Sheng Pai; Her-Terng Yau
2010-01-01
This paper focuses on the chaos control problem of the unified chaotic systems with structured uncertainties. Applying Schur-complement and some matrix manipulation techniques, the controlled uncertain unified chaotic system is then transformed into the linear matrix inequality (LMI) form. Based on Lyapunov stability theory and linear matrix inequality (LMI) formulation, a simple linear feedback control law is obtained to enforce the prespecified exponential decay dynamics of the uncertain un...
Directory of Open Access Journals (Sweden)
S. S. Motsa
2012-01-01
Full Text Available This paper centres on the application of the new piecewise successive linearization method (PSLM in solving the chaotic and nonchaotic Chen system. Numerical simulations are presented graphically and comparison is made between the PSLM and Runge-Kutta-based methods. The work shows that the proposed method provides good accuracy and can be easily extended to other dynamical systems including those that are chaotic in nature.
Complete and generalized synchronization in a class of noise perturbed chaotic systems
Chen, Zhang; Lin, Wei; Zhou, Jie
2007-06-01
In the paper, in light of the LaSalle-type invariance principle for stochastic differential equations, chaos synchronization is investigated for a class of chaotic systems dissatisfying a globally Lipschitz condition with noise perturbation. Sufficient criteria for both complete synchronization and generalized synchronization are rigorously established and thus successfully applied to realize chaos synchronization in the coupled unified chaotic systems. Furthermore, concrete examples as well as their numerical simulations are provided to illustrate the possible application of the established criteria.
An Audio Data Encryption with Single and Double Dimension Discrete-Time Chaotic Systems
AKGÜL, Akif; KAÇAR, Sezgin; Pehlivan, İhsan
2015-01-01
— In this article, a study on increasing security of audio data encryption with single and double dimension discrete-time chaotic systems was carried out and application and security analyses were executed. Audio data samples of both mono and stereo types were encrypted. In the application here, single and double dimension discrete-time chaotic systems were used. In order to enhance security during encryption, a different method was applied by also using a non-linear function. In the chaos ba...
Synchronization of Different Fractional Order Time-Delay Chaotic Systems Using Active Control
Directory of Open Access Journals (Sweden)
Jianeng Tang
2014-01-01
Full Text Available Chaos synchronization of different fractional order time-delay chaotic systems is considered. Based on the Laplace transform theory, the conditions for achieving synchronization of different fractional order time-delay chaotic systems are analyzed by use of active control technique. Then numerical simulations are provided to verify the effectiveness and feasibility of the developed method. At last, effects of the fraction order and the time delay on synchronization are further researched.
Institute of Scientific and Technical Information of China (English)
Zhang Ruo-Xun; Yang Shi-Ping
2008-01-01
This paper presents a general method of the generalized projective synchronization and the parameter identification between two different chaotic systems with unknown parameters.This approach is based on Lyapunov stability theory,and employs a combination of feedback control and adaptive control.With this method one can achieve the generalized projective synchronization and realize the parameter identifications between almost all chaotic (hyperchaotic) systems with unknown parameters.Numerical simulations results are presented to demonstrate the effectiveness of the method.
Synchronization of Different Fractional Order Time-Delay Chaotic Systems Using Active Control
Jianeng Tang
2014-01-01
Chaos synchronization of different fractional order time-delay chaotic systems is considered. Based on the Laplace transform theory, the conditions for achieving synchronization of different fractional order time-delay chaotic systems are analyzed by use of active control technique. Then numerical simulations are provided to verify the effectiveness and feasibility of the developed method. At last, effects of the fraction order and the time delay on synchronization are further researched.
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
The quantum mechanical behavior of classically chaotic systems,usually referred to as quantum chaos,is of interest,since the classical limit is still poorly understood for soft chaos~([1,2]).Here the spatio-temporal evolution of axially symmetric harmonic oscillator coherent states under the action of perturbed Harniltonian with octupole deformation is studied for a classically soft chaotic systems.The initial coherent state
Function Projective Synchronization in Discrete-Time Chaotic System with Uncertain Parameters
Institute of Scientific and Technical Information of China (English)
CHEN Yong; LI Xin
2009-01-01
The function projective synchronization of discrete-time chaotic systems is presented. Based on backstep-ping design with three controllers, a systematic, concrete and automatic scheme is developed to investigate function projective synchronization (FPS) of discrete-time chaotic systems with uncertain parameters. With the aid of symbolic-numeric computation, we use the proposed scheme to illustrate FPS between two identical 3D Henon-like maps with uncertain parameters. Numeric simulations are used to verify the effectiveness of our scheme.
Chaos control in delayed chaotic systems via sliding mode based delayed feedback
Energy Technology Data Exchange (ETDEWEB)
Vasegh, Nastaran [Faculty of Electrical Engineering, K.N. Toosi University of Technology, Seyed Khandan Bridge, Shariati St. 16314, P.O. Box 16315-1355, Tehran (Iran, Islamic Republic of)], E-mail: vasegh@eetd.kntu.ac.ir; Sedigh, Ali Khaki [Faculty of Electrical Engineering, K.N. Toosi University of Technology, Seyed Khandan Bridge, Shariati St. 16314, P.O. Box 16315-1355, Tehran (Iran, Islamic Republic of)
2009-04-15
This paper investigates chaos control for scalar delayed chaotic systems using sliding mode control strategy. Sliding surface design is based on delayed feedback controller. It is shown that the proposed controller can achieve stability for an arbitrary unstable fixed point (UPF) or unstable periodic orbit (UPO) with arbitrary period. The chaotic system used in this study to illustrate the theoretical concepts is the well known Mackey-Glass model. Simulation results show the effectiveness of the designed nonlinear sliding mode controller.
OPEN PROBLEM: Orbits' statistics in chaotic dynamical systems
Arnold, V.
2008-07-01
This paper shows how the measurement of the stochasticity degree of a finite sequence of real numbers, published by Kolmogorov in Italian in a journal of insurances' statistics, can be usefully applied to measure the objective stochasticity degree of sequences, originating from dynamical systems theory and from number theory. Namely, whenever the value of Kolmogorov's stochasticity parameter of a given sequence of numbers is too small (or too big), one may conclude that the conjecture describing this sequence as a sample of independent values of a random variables is highly improbable. Kolmogorov used this strategy fighting (in a paper in 'Doklady', 1940) against Lysenko, who had tried to disprove the classical genetics' law of Mendel experimentally. Calculating his stochasticity parameter value for the numbers from Lysenko's experiment reports, Kolmogorov deduced, that, while these numbers were different from the exact fulfilment of Mendel's 3 : 1 law, any smaller deviation would be a manifestation of the report's number falsification. The calculation of the values of the stochasticity parameter would be useful for many other generators of pseudorandom numbers and for many other chaotically looking statistics, including even the prime numbers distribution (discussed in this paper as an example).