Periodic Solutions of a System of Delay Differential Equations for a Small Delay

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Adu A.M. Wasike

2002-06-01

Full Text Available We prove the existence of an asymptotically stable periodic solution of a system of delay differential equations with a small time delay t > 0. To achieve this, we transform the system of equations into a system of perturbed ordinary differential equations and then use perturbation results to show the existence of an asymptotically stable periodic solution. This approach is contingent on the fact that the system of equations with t = 0 has a stable limit cycle. We also provide a comparative study of the solutions of the original system and the perturbed system.  This comparison lays the ground for proving the existence of periodic solutions of the original system by Schauder's fixed point theorem.

Auto-Bäcklund transformations for a differential-delay equation

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Gordoa, Pilar R.; Pickering, Andrew

2013-03-01

Discrete Painlevé equations have, over recent years, generated much interest. One property of such equations that is considered to be particularly important is the existence of auto-Bäcklund transformations, that is, mappings between solutions of the equation in question, usually involving changes in the values of parameters appearing as coefficients. We have recently presented extensions of discrete Painlevé equations to equations involving derivatives as well as shifts in the independent variable. Here we show how auto-Bäcklund transformations can also be constructed for such differential-delay equations. We emphasise that this is the first time that an auto-Bäcklund transformation has been given for a differential-delay equation.

Lie group classification of first-order delay ordinary differential equations

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Dorodnitsyn, Vladimir A.; Kozlov, Roman; Meleshko, Sergey V.; Winternitz, Pavel

2018-05-01

A group classification of first-order delay ordinary differential equations (DODEs) accompanied by an equation for the delay parameter (delay relation) is presented. A subset of such systems (delay ordinary differential systems or DODSs), which consists of linear DODEs and solution-independent delay relations, have infinite-dimensional symmetry algebras—as do nonlinear ones that are linearizable by an invertible transformation of variables. Genuinely nonlinear DODSs have symmetry algebras of dimension n, . It is shown how exact analytical solutions of invariant DODSs can be obtained using symmetry reduction.

Travelling wave solutions for some time-delayed equations through factorizations

International Nuclear Information System (INIS)

Fahmy, E.S.

2008-01-01

In this work, we use factorization method to find explicit particular travelling wave solutions for the following important nonlinear second-order partial differential equations: The generalized time-delayed Burgers-Huxley, time-delayed convective Fishers, and the generalized time-delayed Burgers-Fisher. Using the particular solutions for these equations we find the general solutions, two-parameter solution, as special cases

ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR A CLASS OF DELAY DIFFERENCE EQUATION

Institute of Scientific and Technical Information of China (English)

ZhuHuiyan; HuangLihong

2005-01-01

We propose a class of delay difference equation with piecewise constant nonlinearity. Such a delay difference equation can be regarded as the discrete analog of a differential equation. The convergence of solutions and the existence of asymptotically stable periodic solutions are investigated for such a class of difference equation.

Modeling of delays in PKPD: classical approaches and a tutorial for delay differential equations.

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Koch, Gilbert; Krzyzanski, Wojciech; Pérez-Ruixo, Juan Jose; Schropp, Johannes

2014-08-01

In pharmacokinetics/pharmacodynamics (PKPD) the measured response is often delayed relative to drug administration, individuals in a population have a certain lifespan until they maturate or the change of biomarkers does not immediately affects the primary endpoint. The classical approach in PKPD is to apply transit compartment models (TCM) based on ordinary differential equations to handle such delays. However, an alternative approach to deal with delays are delay differential equations (DDE). DDEs feature additional flexibility and properties, realize more complex dynamics and can complementary be used together with TCMs. We introduce several delay based PKPD models and investigate mathematical properties of general DDE based models, which serve as subunits in order to build larger PKPD models. Finally, we review current PKPD software with respect to the implementation of DDEs for PKPD analysis.

Analysis of a first-order delay differential-delay equation containing two delays

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Marriott, C.; Vallée, R.; Delisle, C.

1989-09-01

An experimental and numerical analysis of the behavior of a two-delay differential equation is presented. It is shown that much of the system's behavior can be related to the stability behavior of the underlying linearized modes. A new phenomenon, mode crossing, is explored.

Intermittently chaotic oscillations for a differential-delay equation with Gaussian nonlinearity

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Hamilton, Ian

1992-01-01

For a differential-delay equation the time dependence of the variable is a function of the variable at a previous time. We consider a differential-delay equation with Gaussian nonlinearity that displays intermittent chaos. Although not the first example of a differential-delay equation that displays such behavior, for this example the intermittency is classified as type III, and the origin of the intermittent chaos may be qualitatively understood from the limiting forms of the equation for large and small variable magnitudes.

Chaos in the delay logistic equation with discontinuous delays

International Nuclear Information System (INIS)

Sen, Ayan; Mukherjee, Debasis

2009-01-01

This paper analyzes a delay logistic equation which models a feedback control problem. Interval map associated to the system is derived. By calculating Lyapunov exponent, we indicate stable orbit and chaotic phenomenon respectively. The results are verified through computer simulation. We identify the parameter which controls the dynamics.

A simple chaotic delay differential equation

International Nuclear Information System (INIS)

Sprott, J.C.

2007-01-01

The simplest chaotic delay differential equation with a sinusoidal nonlinearity is described, including the route to chaos, Lyapunov exponent spectrum, and chaotic diffusion. It is prototypical of many other high-dimensional chaotic systems

Existence results for impulsive evolution differential equations with state-dependent delay

OpenAIRE

Eduardo Hernandez M.; Rathinasamy Sakthivel; Sueli Tanaka Aki

2008-01-01

We study the existence of mild solution for impulsive evolution abstract differential equations with state-dependent delay. A concrete application to partial delayed differential equations is considered.

Delay chemical master equation: direct and closed-form solutions.

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Leier, Andre; Marquez-Lago, Tatiana T

2015-07-08

The stochastic simulation algorithm (SSA) describes the time evolution of a discrete nonlinear Markov process. This stochastic process has a probability density function that is the solution of a differential equation, commonly known as the chemical master equation (CME) or forward-Kolmogorov equation. In the same way that the CME gives rise to the SSA, and trajectories of the latter are exact with respect to the former, trajectories obtained from a delay SSA are exact representations of the underlying delay CME (DCME). However, in contrast to the CME, no closed-form solutions have so far been derived for any kind of DCME. In this paper, we describe for the first time direct and closed solutions of the DCME for simple reaction schemes, such as a single-delayed unimolecular reaction as well as chemical reactions for transcription and translation with delayed mRNA maturation. We also discuss the conditions that have to be met such that such solutions can be derived.

Perturbations of linear delay differential equations at the verge of instability.

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Lingala, N; Namachchivaya, N Sri

2016-06-01

The characteristic equation for a linear delay differential equation (DDE) has countably infinite roots on the complex plane. This paper considers linear DDEs that are on the verge of instability, i.e., a pair of roots of the characteristic equation lies on the imaginary axis of the complex plane and all other roots have negative real parts. It is shown that when small noise perturbations are present, the probability distribution of the dynamics can be approximated by the probability distribution of a certain one-dimensional stochastic differential equation (SDE) without delay. This is advantageous because equations without delay are easier to simulate and one-dimensional SDEs are analytically tractable. When the perturbations are also linear, it is shown that the stability depends on a specific complex number. The theory is applied to study oscillators with delayed feedback. Some errors in other articles that use multiscale approach are pointed out.

Application of Legendre spectral-collocation method to delay differential and stochastic delay differential equation

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Khan, Sami Ullah; Ali, Ishtiaq

2018-03-01

Explicit solutions to delay differential equation (DDE) and stochastic delay differential equation (SDDE) can rarely be obtained, therefore numerical methods are adopted to solve these DDE and SDDE. While on the other hand due to unstable nature of both DDE and SDDE numerical solutions are also not straight forward and required more attention. In this study, we derive an efficient numerical scheme for DDE and SDDE based on Legendre spectral-collocation method, which proved to be numerical methods that can significantly speed up the computation. The method transforms the given differential equation into a matrix equation by means of Legendre collocation points which correspond to a system of algebraic equations with unknown Legendre coefficients. The efficiency of the proposed method is confirmed by some numerical examples. We found that our numerical technique has a very good agreement with other methods with less computational effort.

Razumikhin-Type Stability Criteria for Differential Equations with Delayed Impulses.

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Wang, Qing; Zhu, Quanxin

2013-01-01

This paper studies stability problems of general impulsive differential equations where time delays occur in both differential and difference equations. Based on the method of Lyapunov functions, Razumikhin technique and mathematical induction, several stability criteria are obtained for differential equations with delayed impulses. Our results show that some systems with delayed impulses may be exponentially stabilized by impulses even if the system matrices are unstable. Some less restrictive sufficient conditions are also given to keep the good stability property of systems subject to certain type of impulsive perturbations. Examples with numerical simulations are discussed to illustrate the theorems. Our results may be applied to complex problems where impulses depend on both current and past states.

Razumikhin-type stability criteria for differential equations with delayed impulses

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Qing Wang

2013-01-01

Full Text Available This paper studies stability problems of general impulsive differential equations where time delays occur in both differential and difference equations. Based on the method of Lyapunov functions, Razumikhin technique and mathematical induction, several stability criteria are obtained for differential equations with delayed impulses. Our results show that some systems with delayed impulses may be exponentially stabilized by impulses even if the system matrices are unstable. Some less restrictive sufficient conditions are also given to keep the good stability property of systems subject to certain type of impulsive perturbations. Examples with numerical simulations are discussed to illustrate the theorems. Our results may be applied to complex problems where impulses depend on both current and past states.

Approximating chaotic saddles for delay differential equations.

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Taylor, S Richard; Campbell, Sue Ann

2007-04-01

Chaotic saddles are unstable invariant sets in the phase space of dynamical systems that exhibit transient chaos. They play a key role in mediating transport processes involving scattering and chaotic transients. Here we present evidence (long chaotic transients and fractal basins of attraction) of transient chaos in a "logistic" delay differential equation. We adapt an existing method (stagger-and-step) to numerically construct the chaotic saddle for this system. This is the first such analysis of transient chaos in an infinite-dimensional dynamical system, and in delay differential equations in particular. Using Poincaré section techniques we illustrate approaches to visualizing the saddle set, and confirm that the saddle has the Cantor-like fractal structure consistent with a chaotic saddle generated by horseshoe-type dynamics.

Approximating chaotic saddles for delay differential equations

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Taylor, S. Richard; Campbell, Sue Ann

2007-04-01

Chaotic saddles are unstable invariant sets in the phase space of dynamical systems that exhibit transient chaos. They play a key role in mediating transport processes involving scattering and chaotic transients. Here we present evidence (long chaotic transients and fractal basins of attraction) of transient chaos in a “logistic” delay differential equation. We adapt an existing method (stagger-and-step) to numerically construct the chaotic saddle for this system. This is the first such analysis of transient chaos in an infinite-dimensional dynamical system, and in delay differential equations in particular. Using Poincaré section techniques we illustrate approaches to visualizing the saddle set, and confirm that the saddle has the Cantor-like fractal structure consistent with a chaotic saddle generated by horseshoe-type dynamics.

Delay-differential equations and the Painlevé transcendents

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Grammaticos, B.; Ramani, A.; Moreira, I. C.

1993-07-01

We apply the recently proposed integrability criterion for differential-difference systems (that blends the classical Painlevé analysis with singularity confinement for discrete systems) to a class of first-order differential-delay equations. Our analysis singles out the family of bi-Riccati equations, as integrability candidates. Among these equations that pass the test some are integrable in a straightforward way (usually by reduction to a standard Riccati equation for some transformed variable) while the remaining ones define new hysterodifferential forms of the Painlevé transcendental equations.

Singular Hopf bifurcation in a differential equation with large state-dependent delay.

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Kozyreff, G; Erneux, T

2014-02-08

We study the onset of sustained oscillations in a classical state-dependent delay (SDD) differential equation inspired by control theory. Owing to the large delays considered, the Hopf bifurcation is singular and the oscillations rapidly acquire a sawtooth profile past the instability threshold. Using asymptotic techniques, we explicitly capture the gradual change from nearly sinusoidal to sawtooth oscillations. The dependence of the delay on the solution can be either linear or nonlinear, with at least quadratic dependence. In the former case, an asymptotic connection is made with the Rayleigh oscillator. In the latter, van der Pol's equation is derived for the small-amplitude oscillations. SDD differential equations are currently the subject of intense research in order to establish or amend general theorems valid for constant-delay differential equation, but explicit analytical construction of solutions are rare. This paper illustrates the use of singular perturbation techniques and the unusual way in which solvability conditions can arise for SDD problems with large delays.

Unconditionally stable difference methods for delay partial differential equations

OpenAIRE

Huang, Chengming; Vandewalle, Stefan

2012-01-01

This paper is concerned with the numerical solution of parabolic partial differential equations with time-delay. We focus in particular on the delay dependent stability analysis of difference methods that use a non-constrained mesh, i.e., the time step-size is not required to be a submultiple of the delay. We prove that the fully discrete system unconditionally preserves the delay dependent asymptotic stability of the linear test problem under consideration, when the following discretizati...

Solving the Linear 1D Thermoelasticity Equations with Pure Delay

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Denys Ya. Khusainov

2015-01-01

Full Text Available We propose a system of partial differential equations with a single constant delay τ>0 describing the behavior of a one-dimensional thermoelastic solid occupying a bounded interval of R1. For an initial-boundary value problem associated with this system, we prove a well-posedness result in a certain topology under appropriate regularity conditions on the data. Further, we show the solution of our delayed model to converge to the solution of the classical equations of thermoelasticity as τ→0. Finally, we deduce an explicit solution representation for the delay problem.

Approximate Method for Solving the Linear Fuzzy Delay Differential Equations

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S. Narayanamoorthy

2015-01-01

Full Text Available We propose an algorithm of the approximate method to solve linear fuzzy delay differential equations using Adomian decomposition method. The detailed algorithm of the approach is provided. The approximate solution is compared with the exact solution to confirm the validity and efficiency of the method to handle linear fuzzy delay differential equation. To show this proper features of this proposed method, numerical example is illustrated.

The Oscillation of a Class of the Fractional-Order Delay Differential Equations

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Qianli Lu

2014-01-01

Full Text Available Several oscillation results are proposed including necessary and sufficient conditions for the oscillation of fractional-order delay differential equations with constant coefficients, the sufficient or necessary and sufficient conditions for the oscillation of fractional-order delay differential equations by analysis method, and the sufficient or necessary and sufficient conditions for the oscillation of delay partial differential equation with three different boundary conditions. For this, α-exponential function which is a kind of functions that play the same role of the classical exponential functions of fractional-order derivatives is used.

Exponential p-stability of impulsive stochastic differential equations with delays

International Nuclear Information System (INIS)

Yang Zhiguo; Xu Daoyi; Xiang Li

2006-01-01

In this Letter, we establish a method to study the exponential p-stability of the zero solution of impulsive stochastic differential equations with delays. By establishing an L-operator inequality and using the properties of M-cone and stochastic analysis technique, we obtain some new conditions ensuring the exponential p-stability of the zero solution of impulsive stochastic differential equations with delays. Two illustrative examples have been provided to show the effectiveness of our results

Numerical Analysis for Stochastic Partial Differential Delay Equations with Jumps

OpenAIRE

Li, Yan; Hu, Junhao

2013-01-01

We investigate the convergence rate of Euler-Maruyama method for a class of stochastic partial differential delay equations driven by both Brownian motion and Poisson point processes. We discretize in space by a Galerkin method and in time by using a stochastic exponential integrator. We generalize some results of Bao et al. (2011) and Jacob et al. (2009) in finite dimensions to a class of stochastic partial differential delay equations with jumps in infinite dimensions.

Local bifurcations in differential equations with state-dependent delay.

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Sieber, Jan

2017-11-01

A common task when analysing dynamical systems is the determination of normal forms near local bifurcations of equilibria. As most of these normal forms have been classified and analysed, finding which particular class of normal form one encounters in a numerical bifurcation study guides follow-up computations. This paper builds on normal form algorithms for equilibria of delay differential equations with constant delay that were developed and implemented in DDE-Biftool recently. We show how one can extend these methods to delay-differential equations with state-dependent delay (sd-DDEs). Since higher degrees of regularity of local center manifolds are still open for sd-DDEs, we give an independent (still only partial) argument which phenomena from the truncated normal must persist in the full sd-DDE. In particular, we show that all invariant manifolds with a sufficient degree of normal hyperbolicity predicted by the normal form exist also in the full sd-DDE.

Local bifurcations in differential equations with state-dependent delay

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Sieber, Jan

2017-11-01

A common task when analysing dynamical systems is the determination of normal forms near local bifurcations of equilibria. As most of these normal forms have been classified and analysed, finding which particular class of normal form one encounters in a numerical bifurcation study guides follow-up computations. This paper builds on normal form algorithms for equilibria of delay differential equations with constant delay that were developed and implemented in DDE-Biftool recently. We show how one can extend these methods to delay-differential equations with state-dependent delay (sd-DDEs). Since higher degrees of regularity of local center manifolds are still open for sd-DDEs, we give an independent (still only partial) argument which phenomena from the truncated normal must persist in the full sd-DDE. In particular, we show that all invariant manifolds with a sufficient degree of normal hyperbolicity predicted by the normal form exist also in the full sd-DDE.

Travelling Solitary Wave Solutions for Generalized Time-delayed Burgers-Fisher Equation

International Nuclear Information System (INIS)

Deng Xijun; Han Libo; Li Xi

2009-01-01

In this paper, travelling wave solutions for the generalized time-delayed Burgers-Fisher equation are studied. By using the first-integral method, which is based on the ring theory of commutative algebra, we obtain a class of travelling solitary wave solutions for the generalized time-delayed Burgers-Fisher equation. A minor error in the previous article is clarified. (general)

Third order differential equations with delay

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Petr Liška

2015-05-01

Full Text Available In this paper, we study the oscillation and asymptotic properties of solutions of certain nonlinear third order differential equations with delay. In particular, we extend results of I. Mojsej (Nonlinear Analysis 68, 2008 and we improve conditions on the property B of N. Parhi and S. Padhi (Indian J. Pure Appl. Math., 33, 2002.

Geometric methods of global attraction in systems of delay differential equations

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El-Morshedy, Hassan A.; Ruiz-Herrera, Alfonso

2017-11-01

In this paper we deduce criteria of global attraction in systems of delay differential equations. Our methodology is new and consists in "dominating" the nonlinear terms of the system by a scalar function and then studying some dynamical properties of that function. One of the crucial benefits of our approach is that we obtain delay-dependent results of global attraction that cover the best delay-independent conditions. We apply our results in a gene regulatory model and the classical Nicholson's blowfly equation with patch structure.

Zero singularities of codimension two and three in delay differential equations

International Nuclear Information System (INIS)

Campbell, Sue Ann; Yuan Yuan

2008-01-01

We give conditions under which a general class of delay differential equations has a point of Bogdanov–Takens or a triple zero bifurcation. We show how a centre manifold projection of the delay equations reduces the dynamics to two- or three-dimensional systems of ordinary differential equations. We put these equations in normal form and determine how the coefficients of the normal forms depend on the original parameters in the model. Finally we apply our results to two neural models and compare the predictions of the theory with numerical bifurcation analysis of the full equations. One model involves a transcritical bifurcation, hence we derive and analyse the appropriate unfoldings for this case

Double Hopf bifurcation in delay differential equations

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Redouane Qesmi

2014-07-01

Full Text Available The paper addresses the computation of elements of double Hopf bifurcation for retarded functional differential equations (FDEs with parameters. We present an efficient method for computing, simultaneously, the coefficients of center manifolds and normal forms, in terms of the original FDEs, associated with the double Hopf singularity up to an arbitrary order. Finally, we apply our results to a nonlinear model with periodic delay. This shows the applicability of the methodology in the study of delay models arising in either natural or technological problems.

Necessary and sufficient conditions for Hopf bifurcation in tri-neuron equation with a delay

International Nuclear Information System (INIS)

Liu Xiaoming; Liao Xiaofeng

2009-01-01

In this paper, we consider the delayed differential equations modeling three-neuron equations with only a time delay. Using the time delay as a bifurcation parameter, necessary and sufficient conditions for Hopf bifurcation to occur are derived. Numerical results indicate that for this model, Hopf bifurcation is likely to occur at suitable delay parameter values.

Nonlinear reaction-diffusion equations with delay: some theorems, test problems, exact and numerical solutions

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Polyanin, A. D.; Sorokin, V. G.

2017-12-01

The paper deals with nonlinear reaction-diffusion equations with one or several delays. We formulate theorems that allow constructing exact solutions for some classes of these equations, which depend on several arbitrary functions. Examples of application of these theorems for obtaining new exact solutions in elementary functions are provided. We state basic principles of construction, selection, and use of test problems for nonlinear partial differential equations with delay. Some test problems which can be suitable for estimating accuracy of approximate analytical and numerical methods of solving reaction-diffusion equations with delay are presented. Some examples of numerical solutions of nonlinear test problems with delay are considered.

Periodic solutions and bifurcations of delay-differential equations

International Nuclear Information System (INIS)

He Jihuan

2005-01-01

In this Letter a simple but effective iteration method is proposed to search for limit cycles or bifurcation curves of delay-differential equations. An example is given to illustrate its convenience and effectiveness

International Conference on Delay Differential and Difference Equations and Applications

CERN Document Server

Pituk, Mihály; Recent Advances in Delay Differential and Difference Equations

2014-01-01

Delay differential and difference equations serve as models for a range of processes in biology, physics, engineering, and control theory. In this volume, the participants of the International Conference on Delay Differential and Difference Equations and Applications, Balatonfüred, Hungary, July 15-19, 2013 present recent research in this quickly-evolving field. The papers relate to the existence, asymptotic, and oscillatory properties of the solutions; stability theory; numerical approximations; and applications to real world phenomena using deterministic and stochastic discrete and continuous dynamical systems.

Numerical Integration of a Class of Singularly Perturbed Delay Differential Equations with Small Shift

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Gemechis File

2012-01-01

Full Text Available We have presented a numerical integration method to solve a class of singularly perturbed delay differential equations with small shift. First, we have replaced the second-order singularly perturbed delay differential equation by an asymptotically equivalent first-order delay differential equation. Then, Simpson’s rule and linear interpolation are employed to get the three-term recurrence relation which is solved easily by discrete invariant imbedding algorithm. The method is demonstrated by implementing it on several linear and nonlinear model examples by taking various values for the delay parameter and the perturbation parameter .

Stability and bifurcation analysis of a generalized scalar delay differential equation.

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Bhalekar, Sachin

2016-08-01

This paper deals with the stability and bifurcation analysis of a general form of equation D(α)x(t)=g(x(t),x(t-τ)) involving the derivative of order α ∈ (0, 1] and a constant delay τ ≥ 0. The stability of equilibrium points is presented in terms of the stability regions and critical surfaces. We provide a necessary condition to exist chaos in the system also. A wide range of delay differential equations involving a constant delay can be analyzed using the results proposed in this paper. The illustrative examples are provided to explain the theory.

Numerical bifurcation analysis of delay differential equations arising from physiological modeling.

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Engelborghs, K; Lemaire, V; Bélair, J; Roose, D

2001-04-01

This paper has a dual purpose. First, we describe numerical methods for continuation and bifurcation analysis of steady state solutions and periodic solutions of systems of delay differential equations with an arbitrary number of fixed, discrete delays. Second, we demonstrate how these methods can be used to obtain insight into complex biological regulatory systems in which interactions occur with time delays: for this, we consider a system of two equations for the plasma glucose and insulin concentrations in a diabetic patient subject to a system of external assistance. The model has two delays: the technological delay of the external system, and the physiological delay of the patient's liver. We compute stability of the steady state solution as a function of two parameters, compare with analytical results and compute several branches of periodic solutions and their stability. These numerical results allow to infer two categories of diabetic patients for which the external system has different efficiency.

Stability criteria for neutral delay differential-algebraic equations

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FAN Ni

2013-10-01

Full Text Available The asymptotic stability of neutral delay differential-algebraic equations is studied in this paper.Two stability criteria described by evaluating a corresponding harmonic function on the boundary of a torus region are presented.

Comparison principle for impulsive functional differential equations with infinite delays and applications

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Li, Xiaodi; Shen, Jianhua; Akca, Haydar; Rakkiyappan, R.

2018-04-01

We introduce the Razumikhin technique to comparison principle and establish some comparison results for impulsive functional differential equations (IFDEs) with infinite delays, where the infinite delays may be infinite time-varying delays or infinite distributed delays. The idea is, under the help of Razumikhin technique, to reduce the study of IFDEs with infinite delays to the study of scalar impulsive differential equations (IDEs) in which the solutions are easy to deal with. Based on the comparison principle, we study the qualitative properties of IFDEs with infinite delays , which include stability, asymptotic stability, exponential stability, practical stability, boundedness, etc. It should be mentioned that the developed results in this paper can be applied to IFDEs with not only infinite delays but also persistent impulsive perturbations. Moreover, even for the special cases of non-impulsive effects or/and finite delays, the criteria prove to be simpler and less conservative than some existing results. Finally, two examples are given to illustrate the effectiveness and advantages of the proposed results.

Delay differential equations recent advances and new directions

CERN Document Server

Balachandran, Balakumar; Gilsinn, David E

2009-01-01

This is a cohesive set of contributions from leading experts on the theory and applications of functional and delay differential equations. The book focuses on theory, symbolic, and numerical methods, which show the practical applications of the concepts.

STABILITY OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY

Institute of Scientific and Technical Information of China (English)

无

2009-01-01

In this paper,we obtain suffcient conditions for the stability in p-th moment of the analytical solutions and the mean square stability of a stochastic differential equation with unbounded delay proposed in [6,10] using the explicit Euler method.

The numerical simulation of convection delayed dominated diffusion equation

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Mohan Kumar P. Murali

2016-01-01

Full Text Available In this paper, we propose a fitted numerical method for solving convection delayed dominated diffusion equation. A fitting factor is introduced and the model equation is discretized by cubic spline method. The error analysis is analyzed for the consider problem. The numerical examples are solved using the present method and compared the result with the exact solution.

Stability, bifurcation and a new chaos in the logistic differential equation with delay

International Nuclear Information System (INIS)

Jiang Minghui; Shen Yi; Jian Jigui; Liao Xiaoxin

2006-01-01

This Letter is concerned with bifurcation and chaos in the logistic delay differential equation with a parameter r. The linear stability of the logistic equation is investigated by analyzing the associated characteristic transcendental equation. Based on the normal form approach and the center manifold theory, the formula for determining the direction of Hopf bifurcation and the stability of bifurcation periodic solution in the first bifurcation values is obtained. By theoretical analysis and numerical simulation, we found a new chaos in the logistic delay differential equation

Analytical determination of the bifurcation thresholds in stochastic differential equations with delayed feedback.

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Gaudreault, Mathieu; Drolet, François; Viñals, Jorge

2010-11-01

Analytical expressions for pitchfork and Hopf bifurcation thresholds are given for a nonlinear stochastic differential delay equation with feedback. Our results assume that the delay time τ is small compared to other characteristic time scales, not a significant limitation close to the bifurcation line. A pitchfork bifurcation line is found, the location of which depends on the conditional average , where x(t) is the dynamical variable. This conditional probability incorporates the combined effect of fluctuation correlations and delayed feedback. We also find a Hopf bifurcation line which is obtained by a multiple scale expansion around the oscillatory solution near threshold. We solve the Fokker-Planck equation associated with the slowly varying amplitudes and use it to determine the threshold location. In both cases, the predicted bifurcation lines are in excellent agreement with a direct numerical integration of the governing equations. Contrary to the known case involving no delayed feedback, we show that the stochastic bifurcation lines are shifted relative to the deterministic limit and hence that the interaction between fluctuation correlations and delay affect the stability of the solutions of the model equation studied.

Integration Processes of Delay Differential Equation Based on Modified Laguerre Functions

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Yeguo Sun

2012-01-01

Full Text Available We propose long-time convergent numerical integration processes for delay differential equations. We first construct an integration process based on modified Laguerre functions. Then we establish its global convergence in certain weighted Sobolev space. The proposed numerical integration processes can also be used for systems of delay differential equations. We also developed a technique for refinement of modified Laguerre-Radau interpolations. Lastly, numerical results demonstrate the spectral accuracy of the proposed method and coincide well with analysis.

OSCILLATION OF IMPULSIVE HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION WITH DELAY

Institute of Scientific and Technical Information of China (English)

无

2006-01-01

In this paper, oscillation properties of the solutions of impulsive hyperbolic equation with delay are investigated via the method of differential inequalities. Sufficient conditions for oscillations of the solutions are established.

Estimation of time- and state-dependent delays and other parameters in functional differential equations

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Murphy, K. A.

1990-01-01

A parameter estimation algorithm is developed which can be used to estimate unknown time- or state-dependent delays and other parameters (e.g., initial condition) appearing within a nonlinear nonautonomous functional differential equation. The original infinite dimensional differential equation is approximated using linear splines, which are allowed to move with the variable delay. The variable delays are approximated using linear splines as well. The approximation scheme produces a system of ordinary differential equations with nice computational properties. The unknown parameters are estimated within the approximating systems by minimizing a least-squares fit-to-data criterion. Convergence theorems are proved for time-dependent delays and state-dependent delays within two classes, which say essentially that fitting the data by using approximations will, in the limit, provide a fit to the data using the original system. Numerical test examples are presented which illustrate the method for all types of delay.

Analytical approximate solutions for a general class of nonlinear delay differential equations.

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Căruntu, Bogdan; Bota, Constantin

2014-01-01

We use the polynomial least squares method (PLSM), which allows us to compute analytical approximate polynomial solutions for a very general class of strongly nonlinear delay differential equations. The method is tested by computing approximate solutions for several applications including the pantograph equations and a nonlinear time-delay model from biology. The accuracy of the method is illustrated by a comparison with approximate solutions previously computed using other methods.

New numerical approximation for solving fractional delay differential equations of variable order using artificial neural networks

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Zúñiga-Aguilar, C. J.; Coronel-Escamilla, A.; Gómez-Aguilar, J. F.; Alvarado-Martínez, V. M.; Romero-Ugalde, H. M.

2018-02-01

In this paper, we approximate the solution of fractional differential equations with delay using a new approach based on artificial neural networks. We consider fractional differential equations of variable order with the Mittag-Leffler kernel in the Liouville-Caputo sense. With this new neural network approach, an approximate solution of the fractional delay differential equation is obtained. Synaptic weights are optimized using the Levenberg-Marquardt algorithm. The neural network effectiveness and applicability were validated by solving different types of fractional delay differential equations, linear systems with delay, nonlinear systems with delay and a system of differential equations, for instance, the Newton-Leipnik oscillator. The solution of the neural network was compared with the analytical solutions and the numerical simulations obtained through the Adams-Bashforth-Moulton method. To show the effectiveness of the proposed neural network, different performance indices were calculated.

Existence and Stability of Traveling Waves for Degenerate Reaction-Diffusion Equation with Time Delay

Science.gov (United States)

Huang, Rui; Jin, Chunhua; Mei, Ming; Yin, Jingxue

2018-01-01

This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction-diffusion equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of c≥c^* for the degenerate reaction-diffusion equation without delay, where c^*>0 is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion equation with small time delay τ >0 . Furthermore, we prove the global existence and uniqueness of C^{α ,β } -solution to the time-delayed degenerate reaction-diffusion equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted L^1 -space. The exponential convergence rate is also derived.

Existence and Stability of Traveling Waves for Degenerate Reaction-Diffusion Equation with Time Delay

Science.gov (United States)

Huang, Rui; Jin, Chunhua; Mei, Ming; Yin, Jingxue

2018-06-01

This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction-diffusion equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of c≥c^* for the degenerate reaction-diffusion equation without delay, where c^*>0 is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion equation with small time delay τ >0. Furthermore, we prove the global existence and uniqueness of C^{α ,β }-solution to the time-delayed degenerate reaction-diffusion equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted L^1-space. The exponential convergence rate is also derived.

Oscillation criteria for third order nonlinear delay differential equations with damping

Directory of Open Access Journals (Sweden)

Said R. Grace

2015-01-01

Full Text Available This note is concerned with the oscillation of third order nonlinear delay differential equations of the form \\[\\label{*} \\left( r_{2}(t\\left( r_{1}(ty^{\\prime}(t\\right^{\\prime}\\right^{\\prime}+p(ty^{\\prime}(t+q(tf(y(g(t=0.\\tag{\\(\\ast\\}\\] In the papers [A. Tiryaki, M. F. Aktas, Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping, J. Math. Anal. Appl. 325 (2007, 54-68] and [M. F. Aktas, A. Tiryaki, A. Zafer, Oscillation criteria for third order nonlinear functional differential equations, Applied Math. Letters 23 (2010, 756-762], the authors established some sufficient conditions which insure that any solution of equation (\\(\\ast\\ oscillates or converges to zero, provided that the second order equation \\[\\left( r_{2}(tz^{\\prime }(t\\right^{\\prime}+\\left(p(t/r_{1}(t\\right z(t=0\\tag{\\(\\ast\\ast\\}\\] is nonoscillatory. Here, we shall improve and unify the results given in the above mentioned papers and present some new sufficient conditions which insure that any solution of equation (\\(\\ast\\ oscillates if equation (\\(\\ast\\ast\\ is nonoscillatory. We also establish results for the oscillation of equation (\\(\\ast\\ when equation (\\(\\ast\\ast\\ is oscillatory.

Study of the dynamics of an equation with two large different-order delays

International Nuclear Information System (INIS)

Kashchenko, I.S.

2016-01-01

The case where the larger delay is proportional to the square of the smaller delay is studied in detail. Regions of stability and instability of the equilibrium state and critical cases are found. In all critical cases, special evolutionary equations (quasinormal forms) are constructed. Their non-local dynamics determines the local behavior of solutions of the original equation [ru

Numerical Hopf bifurcation of Runge-Kutta methods for a class of delay differential equations

International Nuclear Information System (INIS)

Wang Qiubao; Li Dongsong; Liu, M.Z.

2009-01-01

In this paper, we consider the discretization of parameter-dependent delay differential equation of the form y ' (t)=f(y(t),y(t-1),τ),τ≥0,y element of R d . It is shown that if the delay differential equation undergoes a Hopf bifurcation at τ=τ * , then the discrete scheme undergoes a Hopf bifurcation at τ(h)=τ * +O(h p ) for sufficiently small step size h, where p≥1 is the order of the Runge-Kutta method applied. The direction of numerical Hopf bifurcation and stability of bifurcating invariant curve are the same as that of delay differential equation.

Oscillation criteria for fourth-order nonlinear delay dynamic equations

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Yunsong Qi

2013-03-01

Full Text Available We obtain criteria for the oscillation of all solutions to a fourth-order nonlinear delay dynamic equation on a time scale that is unbounded from above. The results obtained are illustrated with examples

A NEW OSCILLATION CRITERION FOR FIRST ORDER NEUTRAL DELAY DIFFERENTIAL EQUATION

Institute of Scientific and Technical Information of China (English)

无

2006-01-01

In this paper, a new sufficient condition for the oscillation of all solutions of first order neutral delay differential equations is obtained. Secondly, the result can also be extended to a general neutral differential equation, and many known results in the literatures are improved.

Asymptotic behavior and stability of second order neutral delay differential equations

NARCIS (Netherlands)

Chen, G.L.; van Gaans, O.W.; Verduyn Lunel, Sjoerd

2014-01-01

We study the asymptotic behavior of a class of second order neutral delay differential equations by both a spectral projection method and an ordinary differential equation method approach. We discuss the relation of these two methods and illustrate some features using examples. Furthermore, a fixed

The two modes extension to the Berk-Breizman equation: Delayed differential equations and asymptotic solutions

International Nuclear Information System (INIS)

Marczynski, Slawomir

2011-01-01

The integro-differential Berk-Breizman (BB) equation, describing the evolution of particle-driven wave mode is transformed into a simple delayed differential equation form ν∂a(τ)/∂τ=a(τ) -a 2 (τ- 1) a(τ- 2). This transformation is also applied to the two modes extension of the BB theory. The obtained solutions are presented together with the derived asymptotic analytical solutions and the numerical results.

Hopf bifurcation formula for first order differential-delay equations

Science.gov (United States)

Rand, Richard; Verdugo, Anael

2007-09-01

This work presents an explicit formula for determining the radius of a limit cycle which is born in a Hopf bifurcation in a class of first order constant coefficient differential-delay equations. The derivation is accomplished using Lindstedt's perturbation method.

Interval oscillation criteria for second-order forced impulsive delay differential equations with damping term.

Science.gov (United States)

Thandapani, Ethiraju; Kannan, Manju; Pinelas, Sandra

2016-01-01

In this paper, we present some sufficient conditions for the oscillation of all solutions of a second order forced impulsive delay differential equation with damping term. Three factors-impulse, delay and damping that affect the interval qualitative properties of solutions of equations are taken into account together. The results obtained in this paper extend and generalize some of the the known results for forced impulsive differential equations. An example is provided to illustrate the main result.

Global dynamics of a nonlocal delayed reaction-diffusion equation on a half plane

Science.gov (United States)

Hu, Wenjie; Duan, Yueliang

2018-04-01

We consider a delayed reaction-diffusion equation with spatial nonlocality on a half plane that describes population dynamics of a two-stage species living in a semi-infinite environment. A Neumann boundary condition is imposed accounting for an isolated domain. To describe the global dynamics, we first establish some a priori estimate for nontrivial solutions after investigating asymptotic properties of the nonlocal delayed effect and the diffusion operator, which enables us to show the permanence of the equation with respect to the compact open topology. We then employ standard dynamical system arguments to establish the global attractivity of the nontrivial equilibrium. The main results are illustrated by the diffusive Nicholson's blowfly equation and the diffusive Mackey-Glass equation.

A note on Burgers' equation with time delay: Instability via finite-time blow-up

International Nuclear Information System (INIS)

Jordan, P.M.

2008-01-01

Burgers' equation with time delay is considered. Using the Cole-Hopf transformation, the exact solution of this nonlinear partial differential equation (PDE) is determined in the context of a (seemingly) well-posed initial-boundary value problem (IBVP) involving homogeneous Dirichlet data. The solution obtained, however, is shown to exhibit a delay-induced instability, suffering blow-up in finite-time

ALMOST AUTOMORPHIC MILD SOLUTIONS TO SOME FRACTIONAL DELAY DIFFERENTIAL EQUATIONS

Institute of Scientific and Technical Information of China (English)

无

2012-01-01

In this paper,a new and general existence and uniqueness theorem of almost automorphic mild solutions is obtained for some fractional delay differential equations,using sectorial operators and the Banach contraction principle.

Iterative method for obtaining the prompt and delayed alpha-modes of the diffusion equation

International Nuclear Information System (INIS)

Singh, K.P.; Degweker, S.B.; Modak, R.S.; Singh, Kanchhi

2011-01-01

Highlights: → A method for obtaining α-modes of the neutron diffusion equation has been developed. → The difference between the prompt and delayed modes is more pronounced for the higher modes. → Prompt and delayed modes differ more in reflector region. - Abstract: Higher modes of the neutron diffusion equation are required in some applications such as second order perturbation theory, and modal kinetics. In an earlier paper we had discussed a method for computing the α-modes of the diffusion equation. The discussion assumed that all neutrons are prompt. The present paper describes an extension of the method for finding the α-modes of diffusion equation with the inclusion of delayed neutrons. Such modes are particularly suitable for expanding the time dependent flux in a reactor for describing transients in a reactor. The method is illustrated by applying it to a three dimensional heavy water reactor model problem. The problem is solved in two and three neutron energy groups and with one and six delayed neutron groups. The results show that while the delayed α-modes are similar to λ-modes they are quite different from prompt modes. The difference gets progressively larger as we go to higher modes.

Reduction of structured population models to threshold-type delay equations and functional differential equations: A case study

Energy Technology Data Exchange (ETDEWEB)

Smith, H.L. (Arizona State Univ., Tempe (United States))

1993-01-01

It is shown by way of a simple example that certain structured population models lead naturally to differential delay equations of the threshold type and that these equations can be transformed in a natural way to functional differential equations. The model examined can be viewed as a model of competition between adults and juveniles of a single population. The results indicate the possibility that this competition leads to instability. 28 refs., 2 figs.

Semigroups on Frechet Spaces and Equations with Infinite Delays

Indian Academy of Sciences (India)

In this paper, we show existence and uniqueness of a solution to a functional differential equation with infinite delay. We choose an appropriate Frechet space so as to cover a large class of functions to be used as initial functions to obtain existence and uniqueness of solutions.

Multivalued stochastic delay differential equations and related ...

African Journals Online (AJOL)

We study the existence and uniqueness of a solution for the multivalued stochastic differential equation with delay (the multivalued term is of subdifferential type):. dX(t) + aφ (X(t))dt ∍ b(t,X(t), Y(t), Z(t)) dt. ⎨ +σ (t, X (t), Y (t), Z (t)) dW (t), t ∈ (s, T). X(t) = ξ (t - s), t ∈ [s - δ, s]. Specify that in this case the coefficients at time t ...

A hybrid approach to parameter identification of linear delay differential equations involving multiple delays

Science.gov (United States)

Marzban, Hamid Reza

2018-05-01

In this paper, we are concerned with the parameter identification of linear time-invariant systems containing multiple delays. The approach is based upon a hybrid of block-pulse functions and Legendre's polynomials. The convergence of the proposed procedure is established and an upper error bound with respect to the L2-norm associated with the hybrid functions is derived. The problem under consideration is first transformed into a system of algebraic equations. The least squares technique is then employed for identification of the desired parameters. Several multi-delay systems of varying complexity are investigated to evaluate the performance and capability of the proposed approximation method. It is shown that the proposed approach is also applicable to a class of nonlinear multi-delay systems. It is demonstrated that the suggested procedure provides accurate results for the desired parameters.

Estimation of delays and other parameters in nonlinear functional differential equations

Science.gov (United States)

Banks, H. T.; Lamm, P. K. D.

1983-01-01

A spline-based approximation scheme for nonlinear nonautonomous delay differential equations is discussed. Convergence results (using dissipative type estimates on the underlying nonlinear operators) are given in the context of parameter estimation problems which include estimation of multiple delays and initial data as well as the usual coefficient-type parameters. A brief summary of some of the related numerical findings is also given.

On solutions of neutral stochastic delay Volterra equations with singular kernels

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Xiaotai Wu

2012-08-01

Full Text Available In this paper, existence, uniqueness and continuity of the adapted solutions for neutral stochastic delay Volterra equations with singular kernels are discussed. In addition, continuous dependence on the initial date is also investigated. Finally, stochastic Volterra equation with the kernel of fractional Brownian motion is studied to illustrate the effectiveness of our results.

On the Controllability of a Differential Equation with Delayed and Advanced Arguments

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Raúl Manzanilla

2010-01-01

Full Text Available A semigroup theory for a differential equation with delayed and advanced arguments is developed, with a detailed description of the infinitesimal generator. This in turn allows to study the exact controllability of the equation, by rewriting it as a classical Cauchy problem.

Shifted Legendre method with residual error estimation for delay linear Fredholm integro-differential equations

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Şuayip Yüzbaşı

2017-03-01

Full Text Available In this paper, we suggest a matrix method for obtaining the approximate solutions of the delay linear Fredholm integro-differential equations with constant coefficients using the shifted Legendre polynomials. The problem is considered with mixed conditions. Using the required matrix operations, the delay linear Fredholm integro-differential equation is transformed into a matrix equation. Additionally, error analysis for the method is presented using the residual function. Illustrative examples are given to demonstrate the efficiency of the method. The results obtained in this study are compared with the known results.

Functional differential equations with unbounded delay in extrapolation spaces

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Mostafa Adimy

2014-08-01

Full Text Available We study the existence, regularity and stability of solutions for nonlinear partial neutral functional differential equations with unbounded delay and a Hille-Yosida operator on a Banach space X. We consider two nonlinear perturbations: the first one is a function taking its values in X and the second one is a function belonging to a space larger than X, an extrapolated space. We use the extrapolation techniques to prove the existence and regularity of solutions and we establish a linearization principle for the stability of the equilibria of our equation.

On Delay-Independent Criteria for Oscillation of Higher-Order Functional Differential Equations

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Yuangong Sun

2011-01-01

Full Text Available We investigate the oscillation of the following higher-order functional differential equation: x(n(t+q(t|x(t-τ|λ-1x(t-τ=e(t, where q(t and e(t are continuous functions on [t0,∞, 1>λ>0 and τ≠0 are constants. Unlike most of delay-dependent oscillation results in the literature, two delay-independent oscillation criteria for the equation are established in both the case τ>0 and the case τ<0 under the assumption that the potentials q(t and e(t change signs on [t0,∞.

Gaussian approximations for stochastic systems with delay: Chemical Langevin equation and application to a Brusselator system

International Nuclear Information System (INIS)

Brett, Tobias; Galla, Tobias

2014-01-01

We present a heuristic derivation of Gaussian approximations for stochastic chemical reaction systems with distributed delay. In particular, we derive the corresponding chemical Langevin equation. Due to the non-Markovian character of the underlying dynamics, these equations are integro-differential equations, and the noise in the Gaussian approximation is coloured. Following on from the chemical Langevin equation, a further reduction leads to the linear-noise approximation. We apply the formalism to a delay variant of the celebrated Brusselator model, and show how it can be used to characterise noise-driven quasi-cycles, as well as noise-triggered spiking. We find surprisingly intricate dependence of the typical frequency of quasi-cycles on the delay period

Gaussian approximations for stochastic systems with delay: chemical Langevin equation and application to a Brusselator system.

Science.gov (United States)

Brett, Tobias; Galla, Tobias

2014-03-28

We present a heuristic derivation of Gaussian approximations for stochastic chemical reaction systems with distributed delay. In particular, we derive the corresponding chemical Langevin equation. Due to the non-Markovian character of the underlying dynamics, these equations are integro-differential equations, and the noise in the Gaussian approximation is coloured. Following on from the chemical Langevin equation, a further reduction leads to the linear-noise approximation. We apply the formalism to a delay variant of the celebrated Brusselator model, and show how it can be used to characterise noise-driven quasi-cycles, as well as noise-triggered spiking. We find surprisingly intricate dependence of the typical frequency of quasi-cycles on the delay period.

Multiscale Analysis of Effects of Additive and Multiplicative Noise on Delay Differential Equations near a Bifurcation Point

International Nuclear Information System (INIS)

Klosek, M.M.

2004-01-01

We study effects of noisy and deterministic perturbations on oscillatory solutions to delay differential equations. We develop the multiscale technique and derive amplitude equations for noisy oscillations near a critical delay. We investigate effects of additive and multiplicative noise. We show that if the magnitudes of noise and deterministic perturbations are balanced, then the oscillatory behavior persists for long times being sustained by the noise. We illustrate the technique and its results on linear and logistic delay equations. (author)

On the method of solution of the differential-delay Toda equation

Science.gov (United States)

Villarroel, Javier; Ablowitz, Mark J.

1993-09-01

The method of solution of the Toda differential-delay equation, which is a reduction of the Toda equation in 2+1 dimensions, is described. An important feature of the solution process is to obtain and study a novel Riemann-Hilbert problem. The latter problem requires factorization across an infinite number of strips with a suitable branching structure. Explicit soliton solutions are given.

Stability Criteria for Differential Equations with Variable Time Delays

Science.gov (United States)

Schley, D.; Shail, R.; Gourley, S. A.

2002-01-01

Time delays are an important aspect of mathematical modelling, but often result in highly complicated equations which are difficult to treat analytically. In this paper it is shown how careful application of certain undergraduate tools such as the Method of Steps and the Principle of the Argument can yield significant results. Certain delay…

INERTIAL MANIFOLDS FOR NONAUTONOMOUS SEMILINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS WITH TIME DELAYS

Institute of Scientific and Technical Information of China (English)

无

2006-01-01

The present paper deals with the long-time behavior of a class of nonautonomous retarded semilinear parabolic differential equations. When the time delays are small enough and the spectral gap conditions hold, the inertial manifolds of the nonautonomous retard parabolic equations are constructed by using the Lyapunov-Perron method.

Stationary distributions of stochastic processes described by a linear neutral delay differential equation

International Nuclear Information System (INIS)

Frank, T D

2005-01-01

Stationary distributions of processes are derived that involve a time delay and are defined by a linear stochastic neutral delay differential equation. The distributions are Gaussian distributions. The variances of the Gaussian distributions are either monotonically increasing or decreasing functions of the time delays. The variances become infinite when fixed points of corresponding deterministic processes become unstable. (letter to the editor)

Oscillation criteria for delay difference equations

Directory of Open Access Journals (Sweden)

Jianhua Shen

2001-01-01

Full Text Available This paper is concerned with the oscillation of all solutions of the delay difference equation $$ x_{n+1}-x_n+p_nx_{n-k}=0, quad n=0,1,2,dots $$ where ${p_n}$ is a sequence of nonnegative real numbers and $k$ is a positive integer. Some new oscillation conditions are established. These conditions concern the case when none of the well-known oscillation conditions $$ limsup_{no infty}sum_{i=0}^kp_{n-i}>1 quad{ m and}quad liminf_{no infty}frac{1}{k}sum_{i=1}^kp_{n-i}>frac{k^k}{(k+1^{k+1}} $$ is satisfied.

OSCILLATION FOR NEUTRAL DELAY DIFFERENTIAL EQUATION WITH POSITIVE AND NEGATIVE COEFFICIENTS

Institute of Scientific and Technical Information of China (English)

无

2006-01-01

In this paper, a new oscillating result is established for the first order neutral delay differential equation with positive and negative coefficients, which improves and generalizes several results in the literatures.

Oscillation Criteria in First Order Neutral Delay Impulsive Differential Equations with Constant Coefficients

International Nuclear Information System (INIS)

Dimitrova, M. B.; Donev, V. I.

2008-01-01

This paper is dealing with the oscillatory properties of first order neutral delay impulsive differential equations and corresponding to them inequalities with constant coefficients. The established sufficient conditions ensure the oscillation of every solution of this type of equations.

Climate models with delay differential equations.

Science.gov (United States)

Keane, Andrew; Krauskopf, Bernd; Postlethwaite, Claire M

2017-11-01

A fundamental challenge in mathematical modelling is to find a model that embodies the essential underlying physics of a system, while at the same time being simple enough to allow for mathematical analysis. Delay differential equations (DDEs) can often assist in this goal because, in some cases, only the delayed effects of complex processes need to be described and not the processes themselves. This is true for some climate systems, whose dynamics are driven in part by delayed feedback loops associated with transport times of mass or energy from one location of the globe to another. The infinite-dimensional nature of DDEs allows them to be sufficiently complex to reproduce realistic dynamics accurately with a small number of variables and parameters. In this paper, we review how DDEs have been used to model climate systems at a conceptual level. Most studies of DDE climate models have focused on gaining insights into either the global energy balance or the fundamental workings of the El Niño Southern Oscillation (ENSO) system. For example, studies of DDEs have led to proposed mechanisms for the interannual oscillations in sea-surface temperature that is characteristic of ENSO, the irregular behaviour that makes ENSO difficult to forecast and the tendency of El Niño events to occur near Christmas. We also discuss the tools used to analyse such DDE models. In particular, the recent development of continuation software for DDEs makes it possible to explore large regions of parameter space in an efficient manner in order to provide a "global picture" of the possible dynamics. We also point out some directions for future research, including the incorporation of non-constant delays, which we believe could improve the descriptive power of DDE climate models.

Climate models with delay differential equations

Science.gov (United States)

Keane, Andrew; Krauskopf, Bernd; Postlethwaite, Claire M.

2017-11-01

A fundamental challenge in mathematical modelling is to find a model that embodies the essential underlying physics of a system, while at the same time being simple enough to allow for mathematical analysis. Delay differential equations (DDEs) can often assist in this goal because, in some cases, only the delayed effects of complex processes need to be described and not the processes themselves. This is true for some climate systems, whose dynamics are driven in part by delayed feedback loops associated with transport times of mass or energy from one location of the globe to another. The infinite-dimensional nature of DDEs allows them to be sufficiently complex to reproduce realistic dynamics accurately with a small number of variables and parameters. In this paper, we review how DDEs have been used to model climate systems at a conceptual level. Most studies of DDE climate models have focused on gaining insights into either the global energy balance or the fundamental workings of the El Niño Southern Oscillation (ENSO) system. For example, studies of DDEs have led to proposed mechanisms for the interannual oscillations in sea-surface temperature that is characteristic of ENSO, the irregular behaviour that makes ENSO difficult to forecast and the tendency of El Niño events to occur near Christmas. We also discuss the tools used to analyse such DDE models. In particular, the recent development of continuation software for DDEs makes it possible to explore large regions of parameter space in an efficient manner in order to provide a "global picture" of the possible dynamics. We also point out some directions for future research, including the incorporation of non-constant delays, which we believe could improve the descriptive power of DDE climate models.

Semigroup Approach to Semilinear Partial Functional Differential Equations with Infinite Delay

Directory of Open Access Journals (Sweden)

Hassane Bouzahir

2007-02-01

Full Text Available We describe a semigroup of abstract semilinear functional differential equations with infinite delay by the use of the Crandall Liggett theorem. We suppose that the linear part is not necessarily densely defined but satisfies the resolvent estimates of the Hille-Yosida theorem. We clarify the properties of the phase space ensuring equivalence between the equation under investigation and the nonlinear semigroup.

Oscillation for equations with positive and negative coefficients and with distributed delay I: General results

Directory of Open Access Journals (Sweden)

Leonid Berezansky

2003-02-01

Full Text Available We study a scalar delay differential equation with a bounded distributed delay, $$ dot{x}(t+ int_{h(t}^t x(s,d_s R(t,s - int_{g(t}^t x(s,d_s T(t,s=0, $$ where $R(t,s$, $T(t,s$ are nonnegative nondecreasing in $s$ for any $t$, $$ R(t,h(t=T(t,g(t=0, quad R(t,s geq T(t,s. $$ We establish a connection between non-oscillation of this differential equation and the corresponding differential inequalities, and between positiveness of the fundamental function and the existence of a nonnegative solution for a nonlinear integral inequality that constructed explicitly. We also present comparison theorems, and explicit non-oscillation and oscillation results. In a separate publication (part II, we will consider applications of this theory to differential equations with several concentrated delays, integrodifferential, and mixed equations.

W-transform for exponential stability of second order delay differential equations without damping terms.

Science.gov (United States)

Domoshnitsky, Alexander; Maghakyan, Abraham; Berezansky, Leonid

2017-01-01

In this paper a method for studying stability of the equation [Formula: see text] not including explicitly the first derivative is proposed. We demonstrate that although the corresponding ordinary differential equation [Formula: see text] is not exponentially stable, the delay equation can be exponentially stable.

Analytical approach for the Floquet theory of delay differential equations.

Science.gov (United States)

Simmendinger, C; Wunderlin, A; Pelster, A

1999-05-01

We present an analytical approach to deal with nonlinear delay differential equations close to instabilities of time periodic reference states. To this end we start with approximately determining such reference states by extending the Poincaré-Lindstedt and the Shohat expansions, which were originally developed for ordinary differential equations. Then we systematically elaborate a linear stability analysis around a time periodic reference state. This allows us to approximately calculate the Floquet eigenvalues and their corresponding eigensolutions by using matrix valued continued fractions.

An accurate solution of point reactor neutron kinetics equations of multi-group of delayed neutrons

International Nuclear Information System (INIS)

Yamoah, S.; Akaho, E.H.K.; Nyarko, B.J.B.

2013-01-01

Highlights: ► Analytical solution is proposed to solve the point reactor kinetics equations (PRKE). ► The method is based on formulating a coefficient matrix of the PRKE. ► The method was applied to solve the PRKE for six groups of delayed neutrons. ► Results shows good agreement with other traditional methods in literature. ► The method is accurate and efficient for solving the point reactor kinetics equations. - Abstract: The understanding of the time-dependent behaviour of the neutron population in a nuclear reactor in response to either a planned or unplanned change in the reactor conditions is of great importance to the safe and reliable operation of the reactor. In this study, an accurate analytical solution of point reactor kinetics equations with multi-group of delayed neutrons for specified reactivity changes is proposed to calculate the change in neutron density. The method is based on formulating a coefficient matrix of the homogenous differential equations of the point reactor kinetics equations and calculating the eigenvalues and the corresponding eigenvectors of the coefficient matrix. A small time interval is chosen within which reactivity relatively stays constant. The analytical method was applied to solve the point reactor kinetics equations with six-groups delayed neutrons for a representative thermal reactor. The problems of step, ramp and temperature feedback reactivities are computed and the results compared with other traditional methods. The comparison shows that the method presented in this study is accurate and efficient for solving the point reactor kinetics equations of multi-group of delayed neutrons

Period doubling phenomenon in a class of time delay equations

International Nuclear Information System (INIS)

Oliveira, C.R. de; Malta, C.P.

1985-01-01

The properties of the solution of a nonlinear time delayed differential equation (infinite dimension) as function of two parameters: the time delay tau and another parameter A (nonlinearity) are investigated. After a Hopf bifurcation period doubling may occur and is characterized by Feigenbaum's delta. A strange atractor is obtained after the period doubling cascade and the largest Lyapunov exponent is calculated indicating that the attractor has low dimension. The behaviour of this Liapunov exponent as function of tau is different from its behaviour as function of A. (Author) [pt

Delay differential equations for mode-locked semiconductor lasers.

Science.gov (United States)

Vladimirov, Andrei G; Turaev, Dmitry; Kozyreff, Gregory

2004-06-01

We propose a new model for passive mode locking that is a set of ordinary delay differential equations. We assume a ring-cavity geometry and Lorentzian spectral filtering of the pulses but do not use small gain and loss and weak saturation approximations. By means of a continuation method, we study mode-locking solutions and their stability. We find that stable mode locking can exist even when the nonlasing state between pulses becomes unstable.

Statistical precision of delayed-neutron nondestructive assay techniques

International Nuclear Information System (INIS)

Bayne, C.K.; McNeany, S.R.

1979-02-01

A theoretical analysis of the statistical precision of delayed-neutron nondestructive assay instruments is presented. Such instruments measure the fissile content of nuclear fuel samples by neutron irradiation and delayed-neutron detection. The precision of these techniques is limited by the statistical nature of the nuclear decay process, but the precision can be optimized by proper selection of system operating parameters. Our method is a three-part analysis. We first present differential--difference equations describing the fundamental physics of the measurements. We then derive and present complete analytical solutions to these equations. Final equations governing the expected number and variance of delayed-neutron counts were computer programmed to calculate the relative statistical precision of specific system operating parameters. Our results show that Poisson statistics do not govern the number of counts accumulated in multiple irradiation-count cycles and that, in general, maximum count precision does not correspond with maximum count as first expected. Covariance between the counts of individual cycles must be considered in determining the optimum number of irradiation-count cycles and the optimum irradiation-to-count time ratio. For the assay system in use at ORNL, covariance effects are small, but for systems with short irradiation-to-count transition times, covariance effects force the optimum number of irradiation-count cycles to be half those giving maximum count. We conclude that the equations governing the expected value and variance of delayed-neutron counts have been derived in closed form. These have been computerized and can be used to select optimum operating parameters for delayed-neutron assay devices

STABILITY THEOREM FOR A CLASS OF PERTURBED NONAUTONOMOUS NEUTRAL DIFFERENTIAL EQUATION WITH UNBOUNDED DELAY

Institute of Scientific and Technical Information of China (English)

无

2008-01-01

In this paper, we consider a one-dimensional nonautonomous neutral differential equation. We obtain sufficient conditions under which the zero solution to this equation with unbounded delay and perturbation is uniformly asymptotically stable.

Alternans promotion in cardiac electrophysiology models by delay differential equations.

Science.gov (United States)

Gomes, Johnny M; Dos Santos, Rodrigo Weber; Cherry, Elizabeth M

2017-09-01

Cardiac electrical alternans is a state of alternation between long and short action potentials and is frequently associated with harmful cardiac conditions. Different dynamic mechanisms can give rise to alternans; however, many cardiac models based on ordinary differential equations are not able to reproduce this phenomenon. A previous study showed that alternans can be induced by the introduction of delay differential equations (DDEs) in the formulations of the ion channel gating variables of a canine myocyte model. The present work demonstrates that this technique is not model-specific by successfully promoting alternans using DDEs for five cardiac electrophysiology models that describe different types of myocytes, with varying degrees of complexity. By analyzing results across the different models, we observe two potential requirements for alternans promotion via DDEs for ionic gates: (i) the gate must have a significant influence on the action potential duration and (ii) a delay must significantly impair the gate's recovery between consecutive action potentials.

Alternans promotion in cardiac electrophysiology models by delay differential equations

Science.gov (United States)

Gomes, Johnny M.; dos Santos, Rodrigo Weber; Cherry, Elizabeth M.

2017-09-01

Cardiac electrical alternans is a state of alternation between long and short action potentials and is frequently associated with harmful cardiac conditions. Different dynamic mechanisms can give rise to alternans; however, many cardiac models based on ordinary differential equations are not able to reproduce this phenomenon. A previous study showed that alternans can be induced by the introduction of delay differential equations (DDEs) in the formulations of the ion channel gating variables of a canine myocyte model. The present work demonstrates that this technique is not model-specific by successfully promoting alternans using DDEs for five cardiac electrophysiology models that describe different types of myocytes, with varying degrees of complexity. By analyzing results across the different models, we observe two potential requirements for alternans promotion via DDEs for ionic gates: (i) the gate must have a significant influence on the action potential duration and (ii) a delay must significantly impair the gate's recovery between consecutive action potentials.

Homotopy analysis solutions of point kinetics equations with one delayed precursor group

International Nuclear Information System (INIS)

Zhu Qian; Luo Lei; Chen Zhiyun; Li Haofeng

2010-01-01

Homotopy analysis method is proposed to obtain series solutions of nonlinear differential equations. Homotopy analysis method was applied for the point kinetics equations with one delayed precursor group. Analytic solutions were obtained using homotopy analysis method, and the algorithm was analysed. The results show that the algorithm computation time and precision agree with the engineering requirements. (authors)

Nonlinear Delay Discrete Inequalities and Their Applications to Volterra Type Difference Equations

Directory of Open Access Journals (Sweden)

Yu Wu

2010-01-01

Full Text Available Delay discrete inequalities with more than one nonlinear term are discussed, which generalize some known results and can be used in the analysis of various problems in the theory of certain classes of discrete equations. Application examples to show boundedness and uniqueness of solutions of a Volterra type difference equation are also given.

Boundary layer phenomena for differential-delay equations with state-dependent time lags: III

Science.gov (United States)

Mallet-Paret, John; Nussbaum, Roger D.

We consider a class of singularly perturbed delay-differential equations of the form ɛ ẋ(t)=f(x(t),x(t-r)), where r= r( x( t)) is a state-dependent delay. We study the asymptotic shape, as ɛ→0, of slowly oscillating periodic solutions. In particular, we show that the limiting shape of such solutions can be explicitly described by the solution of a pair of so-called max-plus equations. We are able thereby to characterize both the regular parts of the solution graph and the internal transition layers arising from the singular perturbation structure.

Travelling wavefronts of a generalized Fisher equation with spatio-temporal delay

International Nuclear Information System (INIS)

Jin Chunhua; Yin Jingxue; Wang Yifu

2009-01-01

We discuss a generalized Fisher equation with a convolution term which introduces a time-delay in the nonlinearity. Special attention is paid to the existence and the asymptotic behavior of travelling wavefronts connecting two uniform steady states.

Controllability of nonlocal second-order impulsive neutral stochastic functional integro-differential equations with delay and Poisson jumps

Directory of Open Access Journals (Sweden)

Diem Dang Huan

2015-12-01

Full Text Available The current paper is concerned with the controllability of nonlocal second-order impulsive neutral stochastic functional integro-differential equations with infinite delay and Poisson jumps in Hilbert spaces. Using the theory of a strongly continuous cosine family of bounded linear operators, stochastic analysis theory and with the help of the Banach fixed point theorem, we derive a new set of sufficient conditions for the controllability of nonlocal second-order impulsive neutral stochastic functional integro-differential equations with infinite delay and Poisson jumps. Finally, an application to the stochastic nonlinear wave equation with infinite delay and Poisson jumps is given.

Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations

International Nuclear Information System (INIS)

Udaltsov, Vladimir S.; Goedgebuer, Jean-Pierre; Larger, Laurent; Cuenot, Jean-Baptiste; Levy, Pascal; Rhodes, William T.

2003-01-01

We report that signal encoding with high-dimensional chaos produced by delayed feedback systems with a strong nonlinearity can be broken. We describe the procedure and illustrate the method with chaotic waveforms obtained from a strongly nonlinear optical system that we used previously to demonstrate signal encryption/decryption with chaos in wavelength. The method can be extended to any systems ruled by nonlinear time-delayed differential equations

Gas-evolution oscillators. 10. A model based on a delay equation

Energy Technology Data Exchange (ETDEWEB)

Bar-Eli, K.; Noyes, R.M. [Univ. of Oregon, Eugene, OR (United States)

1992-09-17

This paper develops a simplified method to model the behavior of a gas-evolution oscillator with two differential delay equations in two unknowns consisting of the population of dissolved molecules in solution and the pressure of the gas.

Gas-evolution oscillators. 10. A model based on a delay equation

International Nuclear Information System (INIS)

Bar-Eli, K.; Noyes, R.M.

1992-01-01

This paper develops a simplified method to model the behavior of a gas-evolution oscillator with two differential delay equations in two unknowns consisting of the population of dissolved molecules in solution and the pressure of the gas

Oscillation of second order neutral dynamic equations with distributed delay

Directory of Open Access Journals (Sweden)

Qiaoshun Yang

2016-06-01

Full Text Available In this paper, we establish new oscillation criteria for second order neutral dynamic equations with distributed delay by employing the generalized Riccati transformation. The obtained theorems essentially improve the oscillation results in the literature. And two examples are provided to illustrate to the versatility of our main results.

One-dimensional unstable eigenfunction and manifold computations in delay differential equations

International Nuclear Information System (INIS)

Green, Kirk; Krauskopf, Bernd; Engelborghs, Koen

2004-01-01

In this paper we present a new numerical technique for computing the unstable eigenfunctions of a saddle periodic orbit in a delay differential equation. This is used to obtain the necessary starting data for an established algorithm for computing one-dimensional (1D) unstable manifolds of an associated saddle fixed point of a suitable Poincare map. To illustrate our method, we investigate an intermittent transition to chaos in a delay system describing a semiconductor laser subject to phase-conjugate feedback

Spectra and pseudospectra of neutral delay differential equations with application to real-time substructuring

NARCIS (Netherlands)

Green, K.; Wagenknecht, T.

2010-01-01

This paper deals with the computation of pseudospectra of neutral delay differential equations (NDDEs) with fixed finite delays. This method provides information on the sensitivity of eigenvalues of the system under perturbations of a given size, allowing one to analyse uncertainties in, for

Stability analysis for a delay differential equations model of a hydraulic turbine speed governor

Science.gov (United States)

Halanay, Andrei; Safta, Carmen A.; Dragoi, Constantin; Piraianu, Vlad F.

2017-01-01

The paper aims to study the dynamic behavior of a speed governor for a hydraulic turbine using a mathematical model. The nonlinear mathematical model proposed consists in a system of delay differential equations (DDE) to be compared with already established mathematical models of ordinary differential equations (ODE). A new kind of nonlinearity is introduced as a time delay. The delays can characterize different running conditions of the speed governor. For example, it is considered that spool displacement of hydraulic amplifier might be blocked due to oil impurities in the oil supply system and so the hydraulic amplifier has a time delay in comparison to the time control. Numerical simulations are presented in a comparative manner. A stability analysis of the hydraulic control system is performed, too. Conclusions of the dynamic behavior using the DDE model of a hydraulic turbine speed governor are useful in modeling and controlling hydropower plants.

Electrocardiogram classification using delay differential equations.

Science.gov (United States)

Lainscsek, Claudia; Sejnowski, Terrence J

2013-06-01

Time series analysis with nonlinear delay differential equations (DDEs) reveals nonlinear as well as spectral properties of the underlying dynamical system. Here, global DDE models were used to analyze 5 min data segments of electrocardiographic (ECG) recordings in order to capture distinguishing features for different heart conditions such as normal heart beat, congestive heart failure, and atrial fibrillation. The number of terms and delays in the model as well as the order of nonlinearity of the model have to be selected that are the most discriminative. The DDE model form that best separates the three classes of data was chosen by exhaustive search up to third order polynomials. Such an approach can provide deep insight into the nature of the data since linear terms of a DDE correspond to the main time-scales in the signal and the nonlinear terms in the DDE are related to nonlinear couplings between the harmonic signal parts. The DDEs were able to detect atrial fibrillation with an accuracy of 72%, congestive heart failure with an accuracy of 88%, and normal heart beat with an accuracy of 97% from 5 min of ECG, a much shorter time interval than required to achieve comparable performance with other methods.

Oscillation of solutions to neutral nonlinear impulsive hyperbolic equations with several delays

Directory of Open Access Journals (Sweden)

Jichen Yang

2013-01-01

Full Text Available In this article, we study oscillatory properties of solutions to neutral nonlinear impulsive hyperbolic partial differential equations with several delays. We establish sufficient conditions for oscillation of all solutions.

A semigroup approach to equations with infinite delay and application to a problem of viscoelasticity

Science.gov (United States)

Renardy, M.

1981-10-01

A semigroup approach to differential-delay equations is developed which seems more suitable for certain partial integro-differential equations than the standard theory. On a formal level, it is demonstrated that the stretching of filaments of viscoelastic liquids can be described by an equation of this form.

Dynamics of one- and two-dimensional fronts in a bistable equation with time-delayed global feedback: Propagation failure and control mechanisms

International Nuclear Information System (INIS)

Boubendir, Yassine; Mendez, Vicenc; Rotstein, Horacio G.

2010-01-01

We study the evolution of fronts in a bistable equation with time-delayed global feedback in the fast reaction and slow diffusion regime. This equation generalizes the Hodgkin-Grafstein and Allen-Cahn equations. We derive a nonlinear equation governing the motion of fronts, which includes a term with delay. In the one-dimensional case this equation is linear. We study the motion of one- and two-dimensional fronts, finding a much richer dynamics than for the previously studied cases (without time-delayed global feedback). We explain the mechanism by which localized fronts created by inhibitory global coupling loose stability in a Hopf bifurcation as the delay time increases. We show that for certain delay times, the prevailing phase is different from that corresponding to the system in the absence of global coupling. Numerical simulations of the partial differential equation are in agreement with the analytical predictions.

Analyses of glass transition phenomena by solving differential equation with delay effect

International Nuclear Information System (INIS)

Takeuchi, A.; Inoue, A.

2007-01-01

A linear differential equation for the analyses of glass transition phenomena has been proposed by taking into account the delay effect due to the change in transportation of atoms near the glass transition temperature (T g ). Under the condition maintaining the order of the differential equation as the second, the non-linear differential equation proposed by Van Den Beukel and Sietsma is modified to obtain the analytic solution for a linear equation by introducing the following points: the delay effect which is described with a term of Mackey-Glass model, a concept of effective free volume (x fe eff ) and its concentration expression (C fe eff ) which correspond to the equilibrium, and an additional term associated with C fe eff . In analyzing the linear equation, Doyle's p-function was used for the integral of reaction rate with respect to temperature (T). It is found that the linear equation proposed in the present study can describe the changes in free volume (x) with increasing temperature in the dx/dT-T chart, the sharp increase in free volume at T g , and over shooting phenomena of free volume slightly above the T g , as experimentally in thermal analyses for metallic glasses. The linear solution obtained in the present study is of great importance for the analyses of the glass transition because the change in free volume with increasing temperature on heating is described with fundamental functions

Parametrically Excited Oscillations of Second-Order Functional Differential Equations and Application to Duffing Equations with Time Delay Feedback

Directory of Open Access Journals (Sweden)

Mervan Pašić

2014-01-01

Full Text Available We study oscillatory behaviour of a large class of second-order functional differential equations with three freedom real nonnegative parameters. According to a new oscillation criterion, we show that if at least one of these three parameters is large enough, then the main equation must be oscillatory. As an application, we study a class of Duffing type quasilinear equations with nonlinear time delayed feedback and their oscillations excited by the control gain parameter or amplitude of forcing term. Finally, some open questions and comments are given for the purpose of further study on this topic.

Some oscillation criteria for the second-order linear delay differential equation

Czech Academy of Sciences Publication Activity Database

Opluštil, Z.; Šremr, Jiří

2011-01-01

Roč. 136, č. 2 (2011), s. 195-204 ISSN 0862-7959 Institutional research plan: CEZ:AV0Z10190503 Keywords : second-order linear differential equation with a delay * oscillatory solution Subject RIV: BA - General Mathematics http://www.dml.cz/handle/10338.dmlcz/141582

Stabilization of the Wave Equation with Boundary Time-Varying Delay

Directory of Open Access Journals (Sweden)

Hao Li

2014-01-01

Full Text Available We study the stabilization of the wave equation with variable coefficients in a bounded domain and a time-varying delay term in the time-varying, weakly nonlinear boundary feedbacks. By the Riemannian geometry methods and a suitable assumption of nonlinearity, we obtain the uniform decay of the energy of the closed loop system.

On time transformations for differential equations with state-dependent delay

Czech Academy of Sciences Publication Activity Database

Rezunenko, Oleksandr

2014-01-01

Roč. 12, č. 2 (2014), s. 298-307 ISSN 1895-1074 R&D Projects: GA ČR GAP103/12/2431 Institutional support: RVO:67985556 Keywords : differential equations * state-dependent delay * time transformations Subject RIV: BD - Theory of Information Impact factor: 0.578, year: 2014 http://library.utia.cas.cz/separaty/2014/AS/rezunenko-0429130.pdf

Stability analysis of a class of fractional delay differential equations

Indian Academy of Sciences (India)

In this paper we analyse stability of nonlinear fractional order delay differential equations of the form D y ( t ) = a f ( y ( t − ) ) − by ( t ) , where D is a Caputo fractional derivative of order 0 < ≤ 1. We describe stability regions using critical curves. To explain the proposed theory, we discuss fractional order logistic ...

About Global Stable of Solutions of Logistic Equation with Delay

Science.gov (United States)

Kaschenko, S. A.; Loginov, D. O.

2017-12-01

The article is devoted to the definition of all the arguments for which all positive solutions of logistic equation with delay tend to zero for t → ∞. The authors have proved the acquainted Wright’s conjecture on evaluation of a multitude of such arguments. An approach that enables subsequent refinement of this evaluation has been developed.

Second-order differential-delay equation to describe a hybrid bistable device

Science.gov (United States)

Vallee, R.; Dubois, P.; Cote, M.; Delisle, C.

1987-08-01

The problem of a dynamical system with delayed feedback, a hybrid bistable device, characterized by n response times and described by an nth-order differential-delay equation (DDE) is discussed. Starting from a linear-stability analysis of the DDE, the effects of the second-order differential terms on the position of the first bifurcation and on the frequency of the resulting self-oscillation are shown. The effects of the third-order differential terms on the first bifurcation are also considered. Experimental results are shown to support the linear analysis.

Dynamics of a delay differential equation model of hepatitis B virus infection.

Science.gov (United States)

Gourley, Stephen A; Kuang, Yang; Nagy, John D

2008-04-01

We formulate and systematically study the global dynamics of a simple model of hepatitis B virus in terms of delay differential equations. This model has two important and novel features compared to the well-known basic virus model in the literature. Specifically, it makes use of the more realistic standard incidence function and explicitly incorporates a time delay in virus production. As a result, the infection reproduction number is no longer dependent on the patient liver size (number of initial healthy liver cells). For this model, the existence and the component values of the endemic steady state are explicitly dependent on the time delay. In certain biologically interesting limiting scenarios, a globally attractive endemic equilibrium can exist regardless of the time delay length.

Stability and oscillation of two coupled Duffing equations with time delay state feedback

International Nuclear Information System (INIS)

El-Bassiouny, A F

2006-01-01

This paper presents an analytical study of the simultaneous principal parametric resonances of two coupled Duffing equations with time delay state feedback. The concept of an equivalent damping related to the delay feedback is proposed and the appropriate choice of the feedback gains and the time delay is discussed from the viewpoint of vibration control. The method of multiple scales is used to determine a set of ordinary differential equations governing the modulation of the amplitudes and phases of the two modes. The first order approximation of the resonances are derived and the effect of time delay on the resonances is investigated. The fixed points correspond to a periodic motion for the starting system and we show the frequency-response curves. We analyse the effect of time delay and the other different parameters on these oscillations. The stability of the fixed points is examined by using the variational method. Numerical solutions are carried out and graphical representations of the results are presented and discussed. Increasing in the time delay τ given decreasing and increasing in the regions of definition and stability respectively and the first mode has decreased magnitudes. The multivalued solutions disappear when decreasing the coefficients of cubic nonlinearities of the second mode α 3 and the detuning parameter σ 2 respectively. Both modes shift to the left for increasing linear feedback gain v 1 and the coefficient of parametric excitation f 1 respectively

Existence of periodic solutions for Rayleigh equations with state-dependent delay

Directory of Open Access Journals (Sweden)

Jehad O. Alzabut

2012-05-01

Full Text Available We establish sufficient conditions for the existence of periodic solutions for a Rayleigh-type equation with state-dependent delay. Our approach is based on the continuation theorem in degree theory, and some analysis techniques. An example illustrates that our approach to this problem is new.

Myshkis type oscillation criteria for second-order linear delay differential equations

Czech Academy of Sciences Publication Activity Database

Opluštil, Z.; Šremr, Jiří

2015-01-01

Roč. 178, č. 1 (2015), s. 143-161 ISSN 0026-9255 Institutional support: RVO:67985840 Keywords : linear second-order delay differential equation * oscillation criteria Subject RIV: BA - General Mathematics Impact factor: 0.664, year: 2015 http://link.springer.com/article/10.1007%2Fs00605-014-0719-y

About positivity of green's functions for nonlocal boundary value problems with impulsive delay equations.

Science.gov (United States)

Domoshnitsky, Alexander; Volinsky, Irina

2014-01-01

The impulsive delay differential equation is considered (Lx)(t) = x'(t) + ∑(i=1)(m) p(i)(t)x(t - τ(i) (t)) = f(t), t ∈ [a, b], x(t j) = β(j)x(t(j - 0)), j = 1,…, k, a = t0 equation are obtained.

Boundary layer phenomena for differential-delay equations with state-dependent time lags, I.

Science.gov (United States)

Mallet-Paret, John; Nussbaum, Roger D.

1992-11-01

In this paper we begin a study of the differential-delay equation \\varepsilon x'(t) = - x(t) + f(x(t - r)), r = r(x(t)) . We prove the existence of periodic solutions for 0equations. In a companion paper these results will be used to investigate the limiting profile and corresponding boundary layer phenomena for periodic solutions as ɛ approaches zero.

OSCILLATION OF A SECOND-ORDER HALF-LINEAR NEUTRAL DAMPED DIFFERENTIAL EQUATION WITH TIME-DELAY

Institute of Scientific and Technical Information of China (English)

无

2012-01-01

In this paper,the oscillation for a class of second-order half-linear neutral damped differential equation with time-delay is studied.By means of Yang-inequality,the generalized Riccati transformation and a certain function,some new sufficient conditions for the oscillation are given for all solutions to the equation.

Kramers-Moyal expansion for stochastic differential equations with single and multiple delays: Applications to financial physics and neurophysics

International Nuclear Information System (INIS)

Frank, T.D.

2007-01-01

We present a generalized Kramers-Moyal expansion for stochastic differential equations with single and multiple delays. In particular, we show that the delay Fokker-Planck equation derived earlier in the literature is a special case of the proposed Kramers-Moyal expansion. Applications for bond pricing and a self-inhibitory neuron model are discussed

System of delay difference equations with continuous time with lag function between two known functions

Directory of Open Access Journals (Sweden)

Hajnalka Péics

2016-08-01

Full Text Available The asymptotic behavior of solutions of the system of difference equations with continuous time and lag function between two known real functions is studied. The cases when the lag function is between two linear delay functions, between two power delay functions and between two constant delay functions are observed and illustrated by examples. The asymptotic estimates of solutions of the considered system are obtained.

Existence and asymptotic stability of a viscoelastic wave equation with a delay

KAUST Repository

Kirane, Mokhtar

2011-07-07

In this paper, we consider the viscoelastic wave equation with a delay term in internal feedbacks; namely, we investigate the following problem, together with initial conditions and boundary conditions of Dirichlet type. Here (x, t) ∈ Ω × (0, ∞), g is a positive real valued decreasing function and μ1, μ2 are positive constants. Under an hypothesis between the weight of the delay term in the feedback and the weight of the term without delay, using the Faedo-Galerkin approximations together with some energy estimates, we prove the global existence of the solutions. Under the same assumptions, general decay results of the energy are established via suitable Lyapunov functionals. © 2011 Springer Basel AG.

Existence and asymptotic stability of a viscoelastic wave equation with a delay

KAUST Repository

Kirane, Mokhtar; Said-Houari, Belkacem

2011-01-01

In this paper, we consider the viscoelastic wave equation with a delay term in internal feedbacks; namely, we investigate the following problem, together with initial conditions and boundary conditions of Dirichlet type. Here (x, t) ∈ Ω × (0, ∞), g is a positive real valued decreasing function and μ1, μ2 are positive constants. Under an hypothesis between the weight of the delay term in the feedback and the weight of the term without delay, using the Faedo-Galerkin approximations together with some energy estimates, we prove the global existence of the solutions. Under the same assumptions, general decay results of the energy are established via suitable Lyapunov functionals. © 2011 Springer Basel AG.

Oscillation Criteria of First Order Neutral Delay Differential Equations with Variable Coefficients

Directory of Open Access Journals (Sweden)

Fatima N. Ahmed

2013-01-01

Full Text Available Some new oscillation criteria are given for first order neutral delay differential equations with variable coefficients. Our results generalize and extend some of the well-known results in the literature. Some examples are considered to illustrate the main results.

Statistical inference for discrete-time samples from affine stochastic delay differential equations

DEFF Research Database (Denmark)

Küchler, Uwe; Sørensen, Michael

2013-01-01

Statistical inference for discrete time observations of an affine stochastic delay differential equation is considered. The main focus is on maximum pseudo-likelihood estimators, which are easy to calculate in practice. A more general class of prediction-based estimating functions is investigated...

EXISTENCE OF SOLUTION TO NONLINEAR SECOND ORDER NEUTRAL STOCHASTIC DIFFERENTIAL EQUATIONS WITH DELAY

Institute of Scientific and Technical Information of China (English)

无

2010-01-01

This paper is concerned with nonlinear second order neutral stochastic differential equations with delay in a Hilbert space. Sufficient conditions for the existence of solution to the system are obtained by Picard iterations.

An existence theorem for a type of functional differential equation with infinite delay

NARCIS (Netherlands)

Izsak, F.

We prove an existence theorem for a functional differential equation with infinite delay using the Schauder fixpoint theorem. We extend a result in [19] applying the fixed point procedure in an appropriate function space.

New stability and boundedness results to Volterra integro-differential equations with delay

Directory of Open Access Journals (Sweden)

Cemil Tunç

2016-04-01

Full Text Available In this paper, we consider a certain non-linear Volterra integro-differential equations with delay. We study stability and boundedness of solutions. The technique of proof involves defining suitable Lyapunov functionals. Our results improve and extend the results obtained in literature.

Stabilization of wave equations with variable coefficient and delay in the dynamical boundary feedback

Directory of Open Access Journals (Sweden)

Dandan Guo

2017-08-01

Full Text Available In this article we consider the boundary stabilization of a wave equation with variable coefficients. This equation has an acceleration term and a delayed velocity term on the boundary. Under suitable geometric conditions, we obtain the exponential decay for the solutions. Our proof relies on the geometric multiplier method and the Lyapunov approach.

Stability analysis of Runge-Kutta methods for nonlinear neutral delay integro-differential equations

Institute of Scientific and Technical Information of China (English)

2007-01-01

The sufficient conditions for the stability and asymptotic stability of Runge-Kutta methods for nonlinear neutral delay integro-differential equations are derived. A numerical test that confirms the theoretical results is given in the end.

Variational Iteration Method for Volterra Functional Integrodifferential Equations with Vanishing Linear Delays

Directory of Open Access Journals (Sweden)

Ali Konuralp

2014-01-01

Full Text Available Application process of variational iteration method is presented in order to solve the Volterra functional integrodifferential equations which have multi terms and vanishing delays where the delay function θ(t vanishes inside the integral limits such that θ(t=qt for 0

On oscillations of solutions to second-order linear delay differential equations

Czech Academy of Sciences Publication Activity Database

Opluštil, Z.; Šremr, Jiří

2013-01-01

Roč. 20, č. 1 (2013), s. 65-94 ISSN 1072-947X Institutional support: RVO:67985840 Keywords : linear second-order delay differential equation * oscillatory solution Subject RIV: BA - General Mathematics Impact factor: 0.340, year: 2013 http://www.degruyter.com/view/j/gmj.2013.20.issue-1/gmj-2013-0001/gmj-2013-0001.xml?format=INT

On oscillations of solutions to second-order linear delay differential equations

Czech Academy of Sciences Publication Activity Database

Opluštil, Z.; Šremr, Jiří

2013-01-01

Roč. 20, č. 1 (2013), s. 65-94 ISSN 1072-947X Institutional support: RVO:67985840 Keywords : linear second-order delay differential equation * oscillatory solution Subject RIV: BA - General Mathematics Impact factor: 0.340, year: 2013 http://www.degruyter.com/view/j/gmj.2013.20.issue-1/gmj-2013-0001/gmj-2013-0001. xml ?format=INT

A mathematical model of a crocodilian population using delay-differential equations.

Science.gov (United States)

Gallegos, Angela; Plummer, Tenecia; Uminsky, David; Vega, Cinthia; Wickman, Clare; Zawoiski, Michael

2008-11-01

The crocodilia have multiple interesting characteristics that affect their population dynamics. They are among several reptile species which exhibit temperature-dependent sex determination (TSD) in which the temperature of egg incubation determines the sex of the hatchlings. Their life parameters, specifically birth and death rates, exhibit strong age-dependence. We develop delay-differential equation (DDE) models describing the evolution of a crocodilian population. In using the delay formulation, we are able to account for both the TSD and the age-dependence of the life parameters while maintaining some analytical tractability. In our single-delay model we also find an equilibrium point and prove its local asymptotic stability. We numerically solve the different models and investigate the effects of multiple delays on the age structure of the population as well as the sex ratio of the population. For all models we obtain very strong agreement with the age structure of crocodilian population data as reported in Smith and Webb (Aust. Wild. Res. 12, 541-554, 1985). We also obtain reasonable values for the sex ratio of the simulated population.

Controlled Nonlinear Stochastic Delay Equations: Part II: Approximations and Pipe-Flow Representations

International Nuclear Information System (INIS)

Kushner, Harold J.

2012-01-01

This is the second part of a work dealing with key issues that have not been addressed in the modeling and numerical optimization of nonlinear stochastic delay systems. We consider new classes of models, such as those with nonlinear functions of several controls (such as products), each with is own delay, controlled random Poisson measure driving terms, admissions control with delayed retrials, and others. Part I was concerned with issues concerning the class of admissible controls and their approximations, since the classical definitions are inadequate for our models. This part is concerned with transportation equation representations and their approximations. Such representations of nonlinear stochastic delay models have been crucial in the development of numerical algorithms with much reduced memory and computational requirements. The representations for the new models are not obvious and are developed. They also provide a template for the adaptation of the Markov chain approximation numerical methods.

Stability with respect to initial time difference for generalized delay differential equations

Directory of Open Access Journals (Sweden)

Ravi Agarwal

2015-02-01

Full Text Available Stability with initial data difference for nonlinear delay differential equations is introduced. This type of stability generalizes the known concept of stability in the literature. It gives us the opportunity to compare the behavior of two nonzero solutions when both initial values and initial intervals are different. Several sufficient conditions for stability and for asymptotic stability with initial time difference are obtained. Lyapunov functions as well as comparison results for scalar ordinary differential equations are employed. Several examples are given to illustrate the theory.

Bifurcation analysis of a delay differential equation model associated with the induction of long-term memory

International Nuclear Information System (INIS)

Hao, Lijie; Yang, Zhuoqin; Lei, Jinzhi

2015-01-01

Highlights: • A delay differentiation equation model for CREB regulation is developed. • Increasing the time delay can generate various bifurcations. • Increasing the time delay can induce chaos by two routes. - Abstract: The ability to form long-term memories is an important function for the nervous system, and the formation process is dynamically regulated through various transcription factors, including CREB proteins. In this paper, we investigate the dynamics of a delay differential equation model for CREB protein activities, which involves two positive and two negative feedbacks in the regulatory network. We discuss the dynamical mechanisms underlying the induction of long-term memory, in which bistability is essential for the formation of long-term memory, while long time delay can destabilize the high level steady state to inhibit the long-term memory formation. The model displays rich dynamical response to stimuli, including monostability, bistability, and oscillations, and can transit between different states by varying the negative feedback strength. Introduction of a time delay to the model can generate various bifurcations such as Hopf bifurcation, fold limit cycle bifurcation, Neimark–Sacker bifurcation of cycles, and period-doubling bifurcation, etc. Increasing the time delay can induce chaos by two routes: quasi-periodic route and period-doubling cascade.

Analytical applications for delayed neutrons

International Nuclear Information System (INIS)

Eccleston, G.W.

1983-01-01

Analytical formulations that describe the time dependence of neutron populations in nuclear materials contain delayed-neutron dependent terms. These terms are important because the delayed neutrons, even though their yields in fission are small, permit control of the fission chain reaction process. Analytical applications that use delayed neutrons range from simple problems that can be solved with the point reactor kinetics equations to complex problems that can only be solved with large codes that couple fluid calculations with the neutron dynamics. Reactor safety codes, such as SIMMER, model transients of the entire reactor core using coupled space-time neutronics and comprehensive thermal-fluid dynamics. Nondestructive delayed-neutron assay instruments are designed and modeled using a three-dimensional continuous-energy Monte Carlo code. Calculations on high-burnup spent fuels and other materials that contain a mix of uranium and plutonium isotopes require accurate and complete information on the delayed-neutron periods, yields, and energy spectra. A continuing need exists for delayed-neutron parameters for all the fissioning isotopes

A dynamic IS-LM business cycle model with two time delays in capital accumulation equation

Science.gov (United States)

Zhou, Lujun; Li, Yaqiong

2009-06-01

In this paper, we analyze a augmented IS-LM business cycle model with the capital accumulation equation that two time delays are considered in investment processes according to Kalecki's idea. Applying stability switch criteria and Hopf bifurcation theory, we prove that time delays cause the equilibrium to lose or gain stability and Hopf bifurcation occurs.

A differential delay equation arising from the sieve of Eratosthenes

Science.gov (United States)

Cheer, A. Y.; Goldston, D. A.

1990-01-01

Consideration is given to the differential delay equation introduced by Buchstab (1937) in connection with an asymptotic formula for the uncanceled terms in the sieve of Eratosthenes. Maier (1985) used this result to show there is unexpected irreqularity in the distribution of primes in short intervals. The function omega(u) is studied in this paper using numerical and analytical techniques. The results are applied to give some numerical constants in Maier's theorem.

Numerical solution of neutral functional-differential equations with proportional delays

Directory of Open Access Journals (Sweden)

Mehmet Giyas Sakar

2017-07-01

Full Text Available In this paper, homotopy analysis method is improved with optimal determination of auxiliary parameter by use of residual error function for solving neutral functional-differential equations (NFDEs with proportional delays. Convergence analysis and error estimate of method are given. Some numerical examples are solved and comparisons are made with the existing results. The numerical results show that the homotopy analysis method with residual error function is very effective and simple.

Existence results for impulsive neutral functional differential equations with state-dependent delay

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Mani Mallika Arjunan

2009-04-01

Full Text Available In this article, we study the existence of mild solutions for a class of impulsive abstract partial neutral functional differential equations with state-dependent delay. The results are obtained by using Leray-Schauder Alternative fixed point theorem. Example is provided to illustrate the main result.

A modified van der Pol equation with delay in a description of the heart action

Directory of Open Access Journals (Sweden)

Zduniak Beata

2014-12-01

Full Text Available In this paper, a modified van der Pol equation is considered as a description of the heart action. This model has a number of interesting properties allowing reconstruction of phenomena observed in physiological experiments as well as in Holter electrocardiographic recordings. Our aim is to study periodic solutions of the modified van der Pol equation and take into consideration the influence of feedback and delay which occur in the normal heart action mode as well as in pathological modes. Usage of certain values for feedback and delay parameters allows simulating the heart action when an accessory conducting pathway is present (Wolff-Parkinson-White syndrome.

A class of quasilinear parabolic equations with infinite delay and application to a problem of viscoelasticity

Science.gov (United States)

Renardy, M.

A semigroup approach to differential-delay equations is developed which reduces such equations to ordinary differential equations on a Banach space of histories and seems more suitable for certain partial integro-differential equations than the standard theory. The method is applied to prove a local-time existence theorem for equations of the form utt = g( uxt, uxt) x, where {∂g}/{∂u xt} > 0 . On a formal level, it is demonstrated that the stretching of filaments of viscoelastic liquids can be described by an equation of this form.

Variational Approaches for the Existence of Multiple Periodic Solutions of Differential Delay Equations

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Rong Cheng

2010-01-01

Full Text Available The existence of multiple periodic solutions of the following differential delay equation (=−((− is established by applying variational approaches directly, where ∈ℝ, ∈(ℝ,ℝ and >0 is a given constant. This means that we do not need to use Kaplan and Yorke's reduction technique to reduce the existence problem of the above equation to an existence problem for a related coupled system. Such a reduction method introduced first by Kaplan and Yorke in (1974 is often employed in previous papers to study the existence of periodic solutions for the above equation and its similar ones by variational approaches.

Oscillation and asymptotic stability of a delay differential equation with Richard's nonlinearity

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Leonid Berezansky

2005-04-01

Full Text Available We obtain sufficient conditions for oscillation of solutions, and for asymptotical stability of the positive equilibrium, of the scalar nonlinear delay differential equation $$ frac{dN}{dt} = r(tN(tBig[a-Big(sum_{k=1}^m b_k N(g_k(tBig^{gamma}Big], $$ where $ g_k(tleq t$.

Analysis of backward differentiation formula for nonlinear differential-algebraic equations with 2 delays.

Science.gov (United States)

Sun, Leping

2016-01-01

This paper is concerned with the backward differential formula or BDF methods for a class of nonlinear 2-delay differential algebraic equations. We obtain two sufficient conditions under which the methods are stable and asymptotically stable. At last, examples show that our methods are true.

A Parameter Robust Method for Singularly Perturbed Delay Differential Equations

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Erdogan Fevzi

2010-01-01

Full Text Available Uniform finite difference methods are constructed via nonstandard finite difference methods for the numerical solution of singularly perturbed quasilinear initial value problem for delay differential equations. A numerical method is constructed for this problem which involves the appropriate Bakhvalov meshes on each time subinterval. The method is shown to be uniformly convergent with respect to the perturbation parameter. A numerical example is solved using the presented method, and the computed result is compared with exact solution of the problem.

Explicit Finite Difference Methods for the Delay Pseudoparabolic Equations

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I. Amirali

2014-01-01

Full Text Available Finite difference technique is applied to numerical solution of the initial-boundary value problem for the semilinear delay Sobolev or pseudoparabolic equation. By the method of integral identities two-level difference scheme is constructed. For the time integration the implicit rule is being used. Based on the method of energy estimates the fully discrete scheme is shown to be absolutely stable and convergent of order two in space and of order one in time. The error estimates are obtained in the discrete norm. Some numerical results confirming the expected behavior of the method are shown.

Controlled Nonlinear Stochastic Delay Equations: Part I: Modeling and Approximations

International Nuclear Information System (INIS)

Kushner, Harold J.

2012-01-01

This two-part paper deals with “foundational” issues that have not been previously considered in the modeling and numerical optimization of nonlinear stochastic delay systems. There are new classes of models, such as those with nonlinear functions of several controls (such as products), each with is own delay, controlled random Poisson measure driving terms, admissions control with delayed retrials, and others. There are two basic and interconnected themes for these models. The first, dealt with in this part, concerns the definition of admissible control. The classical definition of an admissible control as a nonanticipative relaxed control is inadequate for these models and needs to be extended. This is needed for the convergence proofs of numerical approximations for optimal controls as well as to have a well-defined model. It is shown that the new classes of admissible controls do not enlarge the range of the value functions, is closed (together with the associated paths) under weak convergence, and is approximatable by ordinary controls. The second theme, dealt with in Part II, concerns transportation equation representations, and their role in the development of numerical algorithms with much reduced memory and computational requirements.

Delay differential equations via the matrix Lambert W function and bifurcation analysis: application to machine tool chatter.

Science.gov (United States)

Yi, Sun; Nelson, Patrick W; Ulsoy, A Galip

2007-04-01

In a turning process modeled using delay differential equations (DDEs), we investigate the stability of the regenerative machine tool chatter problem. An approach using the matrix Lambert W function for the analytical solution to systems of delay differential equations is applied to this problem and compared with the result obtained using a bifurcation analysis. The Lambert W function, known to be useful for solving scalar first-order DDEs, has recently been extended to a matrix Lambert W function approach to solve systems of DDEs. The essential advantages of the matrix Lambert W approach are not only the similarity to the concept of the state transition matrix in lin ear ordinary differential equations, enabling its use for general classes of linear delay differential equations, but also the observation that we need only the principal branch among an infinite number of roots to determine the stability of a system of DDEs. The bifurcation method combined with Sturm sequences provides an algorithm for determining the stability of DDEs without restrictive geometric analysis. With this approach, one can obtain the critical values of delay, which determine the stability of a system and hence the preferred operating spindle speed without chatter. We apply both the matrix Lambert W function and the bifurcation analysis approach to the problem of chatter stability in turning, and compare the results obtained to existing methods. The two new approaches show excellent accuracy and certain other advantages, when compared to traditional graphical, computational and approximate methods.

A semi-analytical approach for solving of nonlinear systems of functional differential equations with delay

Science.gov (United States)

Rebenda, Josef; Šmarda, Zdeněk

2017-07-01

In the paper, we propose a correct and efficient semi-analytical approach to solve initial value problem for systems of functional differential equations with delay. The idea is to combine the method of steps and differential transformation method (DTM). In the latter, formulas for proportional arguments and nonlinear terms are used. An example of using this technique for a system with constant and proportional delays is presented.

Mono-implicit Runge Kutta schemes for singularly perturbed delay differential equations

Science.gov (United States)

Rihan, Fathalla A.; Al-Salti, Nasser S.

2017-09-01

In this paper, we adapt Mono-Implicit Runge-Kutta schemes for numerical approximations of singularly perturbed delay differential equations. The schemes are developed to reduce the computational cost of the fully implicit method which combine the accuracy of implicit method and efficient implementation. Numerical stability properties of the schemes are investigated. Numerical simulations are provided to show the effectiveness of the method for both stiff and non-stiff initial value problems.

Stable Numerical Approach for Fractional Delay Differential Equations

Science.gov (United States)

Singh, Harendra; Pandey, Rajesh K.; Baleanu, D.

2017-12-01

In this paper, we present a new stable numerical approach based on the operational matrix of integration of Jacobi polynomials for solving fractional delay differential equations (FDDEs). The operational matrix approach converts the FDDE into a system of linear equations, and hence the numerical solution is obtained by solving the linear system. The error analysis of the proposed method is also established. Further, a comparative study of the approximate solutions is provided for the test examples of the FDDE by varying the values of the parameters in the Jacobi polynomials. As in special case, the Jacobi polynomials reduce to the well-known polynomials such as (1) Legendre polynomial, (2) Chebyshev polynomial of second kind, (3) Chebyshev polynomial of third and (4) Chebyshev polynomial of fourth kind respectively. Maximum absolute error and root mean square error are calculated for the illustrated examples and presented in form of tables for the comparison purpose. Numerical stability of the presented method with respect to all four kind of polynomials are discussed. Further, the obtained numerical results are compared with some known methods from the literature and it is observed that obtained results from the proposed method is better than these methods.

On the Asymptotic Properties of Nonlinear Third-Order Neutral Delay Differential Equations with Distributed Deviating Arguments

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Youliang Fu

2016-01-01

Full Text Available This paper is concerned with the asymptotic properties of solutions to a third-order nonlinear neutral delay differential equation with distributed deviating arguments. Several new theorems are obtained which ensure that every solution to this equation either is oscillatory or tends to zero. Two illustrative examples are included.

Solving delay differential equations in S-ADAPT by method of steps.

Science.gov (United States)

Bauer, Robert J; Mo, Gary; Krzyzanski, Wojciech

2013-09-01

S-ADAPT is a version of the ADAPT program that contains additional simulation and optimization abilities such as parametric population analysis. S-ADAPT utilizes LSODA to solve ordinary differential equations (ODEs), an algorithm designed for large dimension non-stiff and stiff problems. However, S-ADAPT does not have a solver for delay differential equations (DDEs). Our objective was to implement in S-ADAPT a DDE solver using the methods of steps. The method of steps allows one to solve virtually any DDE system by transforming it to an ODE system. The solver was validated for scalar linear DDEs with one delay and bolus and infusion inputs for which explicit analytic solutions were derived. Solutions of nonlinear DDE problems coded in S-ADAPT were validated by comparing them with ones obtained by the MATLAB DDE solver dde23. The estimation of parameters was tested on the MATLB simulated population pharmacodynamics data. The comparison of S-ADAPT generated solutions for DDE problems with the explicit solutions as well as MATLAB produced solutions which agreed to at least 7 significant digits. The population parameter estimates from using importance sampling expectation-maximization in S-ADAPT agreed with ones used to generate the data. Published by Elsevier Ireland Ltd.

EXISTENCE OF POSITIVE PERIODIC SOLUTION TO A CLASS OF NONAUTONOMOUS DIFFERENTIAL EQUATION WITH IMPULSES AND DELAY

Institute of Scientific and Technical Information of China (English)

无

2009-01-01

In this paper, by fixed point theorem of Krasnoselskii, we study the positive pe- riodic solution to a class of nonautonomous differential equation with impulses and delay. Firstly, definition of periodic solution and some lemmas are stated. Then some results on the existence of positive periodic solution to the equation are obtained.

Analysis of stability and Hopf bifurcation for a delayed logistic equation

International Nuclear Information System (INIS)

Sun Chengjun; Han Maoan; Lin Yiping

2007-01-01

The dynamics of a logistic equation with discrete delay are investigated, together with the local and global stability of the equilibria. In particular, the conditions under which a sequence of Hopf bifurcations occur at the positive equilibrium are obtained. Explicit algorithm for determining the stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation are derived by using the theory of normal form and center manifold [Hassard B, Kazarino D, Wan Y. Theory and applications of Hopf bifurcation. Cambridge: Cambridge University Press; 1981.]. Global existence of periodic solutions is also established by using a global Hopf bifurcation result of Wu [Symmetric functional differential equations and neural networks with memory. Trans Amer Math Soc 350:1998;4799-38.

Dynamics of second order in time evolution equations with state-dependent delay

Czech Academy of Sciences Publication Activity Database

Chueshov, I.; Rezunenko, Oleksandr

123-124, č. 1 (2015), s. 126-149 ISSN 0362-546X R&D Projects: GA ČR GAP103/12/2431 Institutional support: RVO:67985556 Keywords : Second order evolution equations * State dependent delay * Nonlinear plate * Finite-dimensional attractor Subject RIV: BD - Theory of Information Impact factor: 1.125, year: 2015 http://library.utia.cas.cz/separaty/2015/AS/rezunenko-0444708.pdf

Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay

Czech Academy of Sciences Publication Activity Database

Chueshov, I.; Rezunenko, Oleksandr

2015-01-01

Roč. 14, č. 5 (2015), s. 1685-1704 ISSN 1534-0392 R&D Projects: GA ČR GAP103/12/2431 Institutional support: RVO:67985556 Keywords : Parabolic evolution equations * state-dependent delay * global attractor * finite-dimension * exponential attractor Subject RIV: BC - Control Systems Theory Impact factor: 0.926, year: 2015 http://library.utia.cas.cz/separaty/2015/AS/rezunenko-0444705.pdf

A differential delay equation arising from the sieve of Eratosthenes

Science.gov (United States)

Cheer, A. Y.; Goldston, D. A.

1990-07-01

The differential delay equation defined by ω (u) = 1/u for 1 ≤ u ≤ 2 and (uω (u))' = ω (u - 1) for u ≥ 2 was introduced by Buchstab in connection with an asymptotic formula for the number of uncanceled terms in the sieve of Eratosthenes. Maier has recently used this result to show there is unexpected irregularity in the distribution of primes in short intervals. The function ω (u) is studied in this paper using numerical and analytical techniques. The results are applied to give some numerical constants in Maier's theorem.

Positive Solutions for System of Nonlinear Fractional Differential Equations in Two Dimensions with Delay

Directory of Open Access Journals (Sweden)

Azizollah Babakhani

2010-01-01

Full Text Available We investigate the existence and uniqueness of positive solution for system of nonlinear fractional differential equations in two dimensions with delay. Our analysis relies on a nonlinear alternative of Leray-Schauder type and Krasnoselskii's fixed point theorem in a cone.

Delayed power analysis

International Nuclear Information System (INIS)

Adamovich, L.A.; Azarov, V.V.

1999-01-01

Time dependent core power behavior in a nuclear reactor is described with well-known neutron kinetics equations. At the same time, two portions are distinguished in energy released from uranium nuclei fission; one released directly at fission and another delayed (residual) portion produced during radioactive decay of fission products. While prompt power is definitely described with kinetics equations, the delayed power presentation still remains outstanding. Since in operation the delayed power part is relatively small (about 6%) operation, it can be neglected for small reactivity disturbances assuming that entire power obeys neutron kinetics equations. In case of a high negative reactivity rapidly inserted in core (e.g. reactor scram initiation) the prompt and delayed components can be calculated separately with practically no impact on each other, employing kinetics equations for prompt power and known approximation formulas for delayed portion, named residual in this specific case. Under substantial disturbances the prompt component in the dynamic process becomes commensurable with delayed portion, thus making necessary to take into account their cross impact. A system of differential equations to describe time-dependent behavior of delayed power is presented. Specific NPP analysis shows a way to significantly simplify the task formulation. (author)

A model of a fishery with fish stock involving delay equations.

Science.gov (United States)

Auger, P; Ducrot, Arnaud

2009-12-13

The aim of this paper is to provide a new mathematical model for a fishery by including a stock variable for the resource. This model takes the form of an infinite delay differential equation. It is mathematically studied and a bifurcation analysis of the steady states is fulfilled. Depending on the different parameters of the problem, we show that Hopf bifurcation may occur leading to oscillating behaviours of the system. The mathematical results are finally discussed.

Delay Mitigation in the Malaysian Housing Industry: A Structural Equation Modelling Approach

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Chang Saar Chai

2015-01-01

Full Text Available The housing industry is one of the major contributors to the economy in Malaysia due to the constantly high housing demand. The housing demand has increased due to the rapid growth in population and urbanisation in the country. One of the major challenges in the housing industry is the late delivery of housing supply, which in some instances leads to sick and abandoned housing projects. Despite being extensively investigated, th in a negative impact, there is a strong need to review the housing delay mitigation measures practised in Malaysia. This paper aims to evaluate the current delay mitigation measures and its main objective is to explore the relationship between the mitigation measures and delay in housing via a Structural Equation Modelling (SEM approach. A questionnaire survey through an online survey tool was conducted across 13 states and three Federal Territories in Malaysia. The target respondents are the local authorities, developers, consultants (principal submitting persons and contractors. The findings show that 17 predictive, preventive, organisational or corrective. This paper demonstrates that preventive measures are the most influential mitigation measures for housing delivery delay.

Some properties of solutions of a functional-differential equation of second order with delay.

Science.gov (United States)

Ilea, Veronica Ana; Otrocol, Diana

2014-01-01

Existence, uniqueness, data dependence (monotony, continuity, and differentiability with respect to parameter), and Ulam-Hyers stability results for the solutions of a system of functional-differential equations with delays are proved. The techniques used are Perov's fixed point theorem and weakly Picard operator theory.

Appling Laplace Adomian decomposition method for delay differential equations with boundary value problems

Science.gov (United States)

Yousef, Hamood Mohammed; Ismail, Ahmad Izani

2017-11-01

In this paper, Laplace Adomian decomposition method (LADM) was applied to solve Delay differential equations with Boundary Value Problems. The solution is in the form of a convergent series which is easy to compute. This approach is tested on two test problem. The findings obtained exhibit the reliability and efficiency of the proposed method.

Some Properties of Solutions of a Functional-Differential Equation of Second Order with Delay

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Veronica Ana Ilea

2014-01-01

Full Text Available Existence, uniqueness, data dependence (monotony, continuity, and differentiability with respect to parameter, and Ulam-Hyers stability results for the solutions of a system of functional-differential equations with delays are proved. The techniques used are Perov’s fixed point theorem and weakly Picard operator theory.

Almost-Periodic Weak Solutions of Second-Order Neutral Delay-Differential Equations with Piecewise Constant Argument

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Wang Li

2008-01-01

Full Text Available We investigate the existence of almost-periodic weak solutions of second-order neutral delay-differential equations with piecewise constant argument of the form , where denotes the greatest integer function, is a real nonzero constant, and is almost periodic.

Response of an oscillatory differential delay equation to a single stimulus.

Science.gov (United States)

Mackey, Michael C; Tyran-Kamińska, Marta; Walther, Hans-Otto

2017-04-01

Here we analytically examine the response of a limit cycle solution to a simple differential delay equation to a single pulse perturbation of the piecewise linear nonlinearity. We construct the unperturbed limit cycle analytically, and are able to completely characterize the perturbed response to a pulse of positive amplitude and duration with onset at different points in the limit cycle. We determine the perturbed minima and maxima and period of the limit cycle and show how the pulse modifies these from the unperturbed case.

Derivation and computation of discrete-delay and continuous-delay SDEs in mathematical biology.

Science.gov (United States)

Allen, Edward J

2014-06-01

Stochastic versions of several discrete-delay and continuous-delay differential equations, useful in mathematical biology, are derived from basic principles carefully taking into account the demographic, environmental, or physiological randomness in the dynamic processes. In particular, stochastic delay differential equation (SDDE) models are derived and studied for Nicholson's blowflies equation, Hutchinson's equation, an SIS epidemic model with delay, bacteria/phage dynamics, and glucose/insulin levels. Computational methods for approximating the SDDE models are described. Comparisons between computational solutions of the SDDEs and independently formulated Monte Carlo calculations support the accuracy of the derivations and of the computational methods.

Multiple periodic solutions for a class of second-order nonlinear neutral delay equations

Directory of Open Access Journals (Sweden)

2006-01-01

Full Text Available By means of a variational structure and Z 2 -group index theory, we obtain multiple periodic solutions to a class of second-order nonlinear neutral delay equations of the form0, au>0$"> x ″ ( t − τ + λ ( t f ( t , x ( t , x ( t − τ , x ( t − 2 τ = x ( t , λ ( t > 0 , τ > 0 .

Stationary solutions of linear stochastic delay differential equations: applications to biological systems.

Science.gov (United States)

Frank, T D; Beek, P J

2001-08-01

Recently, Küchler and Mensch [Stochastics Stochastics Rep. 40, 23 (1992)] derived exact stationary probability densities for linear stochastic delay differential equations. This paper presents an alternative derivation of these solutions by means of the Fokker-Planck approach introduced by Guillouzic [Phys. Rev. E 59, 3970 (1999); 61, 4906 (2000)]. Applications of this approach, which is argued to have greater generality, are discussed in the context of stochastic models for population growth and tracking movements.

Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations

NARCIS (Netherlands)

Jarlebring, E.; Hochstenbach, M.E.

2009-01-01

Several recent methods used to analyze asymptotic stability of delay-differential equations (DDEs) involve determining the eigenvalues of a matrix, a matrix pencil or a matrix polynomial constructed by Kronecker products. Despite some similarities between the different types of these so-called

Existence of Solution of Neutral Fractional Impulsive Differential Equations with Infinite Delay

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K.Hilal

2017-04-01

Full Text Available In this work we define the solution of the fractional neutral impulsive differential equation with infinite delay. The results of existence are obtained by using the Banach contraction and Schafer fixed point. Some proprieties of probability density functions and semi-group theory are also used.

A modified van der Pol equation with delay in a description of the heart action

OpenAIRE

Zduniak Beata; Bodnar Marek; Foryś Urszula

2014-01-01

In this paper, a modified van der Pol equation is considered as a description of the heart action. This model has a number of interesting properties allowing reconstruction of phenomena observed in physiological experiments as well as in Holter electrocardiographic recordings. Our aim is to study periodic solutions of the modified van der Pol equation and take into consideration the influence of feedback and delay which occur in the normal heart action mode as well as in pathological modes. U...

Analytical estimations of limit cycle amplitude for delay-differential equations

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Tamás Molnár

2016-09-01

Full Text Available The amplitude of limit cycles arising from Hopf bifurcation is estimated for nonlinear delay-differential equations by means of analytical formulas. An improved analytical estimation is introduced, which allows more accurate quantitative prediction of periodic solutions than the standard approach that formulates the amplitude as a square-root function of the bifurcation parameter. The improved estimation is based on special global properties of the system: the method can be applied if the limit cycle blows up and disappears at a certain value of the bifurcation parameter. As an illustrative example, the improved analytical formula is applied to the problem of stick balancing.

Noise-induced transitions at a Hopf bifurcation in a first-order delay-differential equation

International Nuclear Information System (INIS)

Longtin, A.

1991-01-01

The influence of colored noise on the Hopf bifurcation in a first-order delay-differential equation (DDE), a model paradigm for nonlinear delayed feedback systems, is considered. First, it is shown, using a stability analysis, how the properties of the DDE depend on the ratio R of system delay to response time. When this ratio is small, the DDE behaves more like a low-dimensional system of ordinary differential equations (ODE's); when R is large, one obtains a singular perturbation limit in which the behavior of the DDE approaches that of a discrete time map. The relative magnitude of the additive and multiplicative noise-induced postponements of the Hopf bifurcation are numerically shown to depend on the ratio R. Although both types of postponements are minute in the large-R limit, they are almost equal due to an equivalence of additive and parametric noise for discrete time maps. When R is small, the multiplicative shift is larger than the additive one at large correlation times, but the shifts are equal for small correlation times. In fact, at constant noise power, the postponement is only slightly affected by the correlation time of the noise, except when the noise becomes white, in which case the postponement drastically decreases. This is a numerical study of the stochastic Hopf bifurcation, in ODE's or DDE's, that looks at the effect of noise correlation time at constant power. Further, it is found that the slope at the fixed point averaged over the stochastic-parameter motion acts, under certain conditions, as a quantitative indicator of oscillation onset in the presence of noise. The problem of how properties of the DDE carry over to ODE's and to maps is discussed, along with the proper theoretical framework in which to study nonequilibrium phase transitions in this class of functional differential equations

A delay differential equation model of follicle waves in women.

Science.gov (United States)

Panza, Nicole M; Wright, Andrew A; Selgrade, James F

2016-01-01

This article presents a mathematical model for hormonal regulation of the menstrual cycle which predicts the occurrence of follicle waves in normally cycling women. Several follicles of ovulatory size that develop sequentially during one menstrual cycle are referred to as follicle waves. The model consists of 13 nonlinear, delay differential equations with 51 parameters. Model simulations exhibit a unique stable periodic cycle and this menstrual cycle accurately approximates blood levels of ovarian and pituitary hormones found in the biological literature. Numerical experiments illustrate that the number of follicle waves corresponds to the number of rises in pituitary follicle stimulating hormone. Modifications of the model equations result in simulations which predict the possibility of two ovulations at different times during the same menstrual cycle and, hence, the occurrence of dizygotic twins via a phenomenon referred to as superfecundation. Sensitive parameters are identified and bifurcations in model behaviour with respect to parameter changes are discussed. Studying follicle waves may be helpful for improving female fertility and for understanding some aspects of female reproductive ageing.

A fixed point approach towards stability of delay differential equations with applications to neural networks

NARCIS (Netherlands)

Chen, Guiling

2013-01-01

This thesis studies asymptotic behavior and stability of determinsitic and stochastic delay differential equations. The approach used in this thesis is based on fixed point theory, which does not resort to any Liapunov function or Liapunov functional. The main contribution of this thesis is to study

Interval Oscillation Criteria of Second Order Mixed Nonlinear Impulsive Differential Equations with Delay

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Zhonghai Guo

2012-01-01

Full Text Available We study the following second order mixed nonlinear impulsive differential equations with delay (r(tΦα(x′(t′+p0(tΦα(x(t+∑i=1npi(tΦβi(x(t-σ=e(t, t≥t0, t≠τk,x(τk+=akx(τk, x'(τk+=bkx'(τk, k=1,2,…, where Φ*(u=|u|*-1u, σ is a nonnegative constant, {τk} denotes the impulsive moments sequence, and τk+1-τk>σ. Some sufficient conditions for the interval oscillation criteria of the equations are obtained. The results obtained generalize and improve earlier ones. Two examples are considered to illustrate the main results.

Two-color pump-probe laser spectroscopy instrument with picosecond time-resolved electronic delay and extended scan range

Science.gov (United States)

Yu, Anchi; Ye, Xiong; Ionascu, Dan; Cao, Wenxiang; Champion, Paul M.

2005-11-01

An electronically delayed two-color pump-probe instrument was developed using two synchronized laser systems. The instrument has picosecond time resolution and can perform scans over hundreds of nanoseconds without the beam divergence and walk-off effects that occur using standard spatial delay systems. A unique picosecond Ti :sapphire regenerative amplifier was also constructed without the need for pulse stretching and compressing optics. The picosecond regenerative amplifier has a broad wavelength tuning range, which suggests that it will make a significant contribution to two-color pump-probe experiments. To test this instrument we studied the rotational correlation relaxation of myoglobin (τr=8.2±0.5ns) in water as well as the geminate rebinding kinetics of oxygen to myoglobin (kg1=1.7×1011s-1, kg2=3.4×107s-1). The results are consistent with, and improve upon, previous studies.

Delay differential equations and the dose-time dependence of early radiotherapy reactions

International Nuclear Information System (INIS)

Fenwick, John D.

2006-01-01

The dose-time dependence of early radiotherapy reactions impacts on the design of accelerated fractionation schedules--oral mucositis, for example, can be dose limiting for short treatments designed to avoid tumor repopulation. In this paper a framework for modeling early reaction dose-time dependence is developed. Variation of stem cell number with time after the start of a radiation schedule is modeled using a first-order delay differential equation (DDE), motivated by experimental observations linking the speed of compensatory proliferation in early reacting tissues to the degree of tissue damage. The modeling suggests that two types of early reaction radiation response are possible, stem cell numbers either monotonically approaching equilibrium plateau levels or overshooting before returning to equilibrium. Several formulas have been derived from the delay differential equation, predicting changes in isoeffective total radiation dose with schedule duration for different types of fractionation scheme. The formulas have been fitted to a wide range of published animal early reaction data, the fits all implying a degree of overshoot. Results are presented illustrating the scope of the delay differential model: most of the data are fitted well, although the model struggles with a few datasets measured for schedules with distinctive dose-time patterns. Ways of extending the current model to cope with these particular dose-time patterns are briefly discussed. The DDE approach is conceptually more complex than earlier descriptive dose-time models but potentially more powerful. It can be used to study issues not addressed by simpler models, such as the likely effects of increasing or decreasing the dose-per-day over time, or of splitting radiation courses into intense segments separated by gaps. It may also prove useful for modeling the effects of chemoirradiation

Delay differential equations and the dose-time dependence of early radiotherapy reactions.

Science.gov (United States)

Fenwick, John D

2006-09-01

The dose-time dependence of early radiotherapy reactions impacts on the design of accelerated fractionation schedules--oral mucositis, for example, can be dose limiting for short treatments designed to avoid tumor repopulation. In this paper a framework for modeling early reaction dose-time dependence is developed. Variation of stem cell number with time after the start of a radiation schedule is modeled using a first-order delay differential equation (DDE), motivated by experimental observations linking the speed of compensatory proliferation in early reacting tissues to the degree of tissue damage. The modeling suggests that two types of early reaction radiation response are possible, stem cell numbers either monotonically approaching equilibrium plateau levels or overshooting before returning to equilibrium. Several formulas have been derived from the delay differential equation, predicting changes in isoeffective total radiation dose with schedule duration for different types of fractionation scheme. The formulas have been fitted to a wide range of published animal early reaction data, the fits all implying a degree of overshoot. Results are presented illustrating the scope of the delay differential model: most of the data are fitted well, although the model struggles with a few datasets measured for schedules with distinctive dose-time patterns. Ways of extending the current model to cope with these particular dose-time patterns are briefly discussed. The DDE approach is conceptually more complex than earlier descriptive dose-time models but potentially more powerful. It can be used to study issues not addressed by simpler models, such as the likely effects of increasing or decreasing the dose-per-day over time, or of splitting radiation courses into intense segments separated by gaps. It may also prove useful for modeling the effects of chemoirradiation.

Parabolic partial differential equations with discrete state-dependent delay: Classical solutions and solution manifold

Czech Academy of Sciences Publication Activity Database

Krisztin, T.; Rezunenko, Oleksandr

2016-01-01

Roč. 260, č. 5 (2016), s. 4454-4472 ISSN 0022-0396 R&D Projects: GA ČR GAP103/12/2431 Institutional support: RVO:67985556 Keywords : Parabolic partial differential equations * State dependent delay * Solution manifold Subject RIV: BC - Control Systems Theory Impact factor: 1.988, year: 2016 http://library.utia.cas.cz/separaty/2016/AS/rezunenko-0457879.pdf

Explosive attractor solutions to a universal cubic delay equation

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Sanz-Orozco, D.; Berk, H. L.

2017-05-01

New explosive attractor solutions have been found in a universal cubic delay equation that has been studied in both the plasma and the fluid mechanics literature. Through computational simulations and analytic approximations, it is found that the oscillatory component of the explosive mode amplitude solutions are described through multi-frequency Fourier expansions with respect to a pseudo-time variable. The spectral dependence of these solutions as a function of a system parameter, ϕ , is studied. The mode amplitude that is described in the explosive regime has two main features: a well-known envelope ( t 0 - t ) - 5 / 2 , with t0 the blow-up time of the amplitude, and a spectrum of discrete oscillations with ever-increasing frequencies, which may give experimental information about the properties of a system's equilibrium.

Variation formulae for the solutions of delay differential equations with discontinuous initial conditions

International Nuclear Information System (INIS)

Kharatishvili, G L; Tadumadze, T A

2005-01-01

Variation formulae are proved for solutions of non-linear differential equations with variable delays and discontinuous initial conditions. The discontinuity of the initial condition means that at the initial moment of time the values of the initial function and the trajectory, generally speaking, do not coincide. The formulae obtained contain a new summand connected with the discontinuity of the initial condition and the variation of the initial moment.

About sign-constancy of Green's functions for impulsive second order delay equations

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Alexander Domoshnitsky

2014-01-01

Full Text Available We consider the following second order differential equation with delay \\[\\begin{cases} (Lx(t\\equiv{x''(t+\\sum_{j=1}^p {b_{j}(tx(t-\\theta_{j}(t}}=f(t, \\quad t\\in[0,\\omega],\\\\ x(t_j=\\gamma_{j}x(t_j-0, x'(t_j=\\delta_{j}x'(t_j-0, \\quad j=1,2,\\ldots,r. \\end{cases}\\] In this paper we find necessary and sufficient conditions of positivity of Green's functions for this impulsive equation coupled with one or two-point boundary conditions in the form of theorems about differential inequalities. By choosing the test function in these theorems, we obtain simple sufficient conditions. For example, the inequality \\(\\sum_{i=1}^p{b_i(t\\left(\\frac{1}{4}+r\\right}\\lt \\frac{2}{\\omega^2}\\ is a basic one, implying negativity of Green's function of two-point problem for this impulsive equation in the case \\(0\\lt \\gamma_i\\leq{1}\\, \\(0\\lt \\delta_i\\leq{1}\\ for \\(i=1,\\ldots ,p\\.

INTERVAL STATE ESTIMATION FOR SINGULAR DIFFERENTIAL EQUATION SYSTEMS WITH DELAYS

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T. A. Kharkovskaia

2016-07-01

Full Text Available The paper deals with linear differential equation systems with algebraic restrictions (singular systems and a method of interval observer design for this kind of systems. The systems contain constant time delay, measurement noise and disturbances. Interval observer synthesis is based on monotone and cooperative systems technique, linear matrix inequations, Lyapunov function theory and interval arithmetic. The set of conditions that gives the possibility for interval observer synthesis is proposed. Results of synthesized observer operation are shown on the example of dynamical interindustry balance model. The advantages of proposed method are that it is adapted to observer design for uncertain systems, if the intervals of admissible values for uncertain parameters are given. The designed observer is capable to provide asymptotically definite limits on the estimation accuracy, since the interval of admissible values for the object state is defined at every instant. The obtained result provides an opportunity to develop the interval estimation theory for complex systems that contain parametric uncertainty, varying delay and nonlinear elements. Interval observers increasingly find applications in economics, electrical engineering, mechanical systems with constraints and optimal flow control.

Analytic Approximation of the Solutions of Stochastic Differential Delay Equations with Poisson Jump and Markovian Switching

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Hua Yang

2012-01-01

Full Text Available We are concerned with the stochastic differential delay equations with Poisson jump and Markovian switching (SDDEsPJMSs. Most SDDEsPJMSs cannot be solved explicitly as stochastic differential equations. Therefore, numerical solutions have become an important issue in the study of SDDEsPJMSs. The key contribution of this paper is to investigate the strong convergence between the true solutions and the numerical solutions to SDDEsPJMSs when the drift and diffusion coefficients are Taylor approximations.

The Use of Generalized Laguerre Polynomials in Spectral Methods for Solving Fractional Delay Differential Equations.

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Khader, M M

2013-10-01

In this paper, an efficient numerical method for solving the fractional delay differential equations (FDDEs) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based on the derived approximate formula of the Laguerre polynomials. The properties of Laguerre polynomials are utilized to reduce FDDEs to a linear or nonlinear system of algebraic equations. Special attention is given to study the error and the convergence analysis of the proposed method. Several numerical examples are provided to confirm that the proposed method is in excellent agreement with the exact solution.

2–stage stochastic Runge–Kutta for stochastic delay differential equations

Energy Technology Data Exchange (ETDEWEB)

Rosli, Norhayati; Jusoh Awang, Rahimah [Faculty of Industrial Science and Technology, Universiti Malaysia Pahang, Lebuhraya Tun Razak, 26300, Gambang, Pahang (Malaysia); Bahar, Arifah; Yeak, S. H. [Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor (Malaysia)

2015-05-15

This paper proposes a newly developed one-step derivative-free method, that is 2-stage stochastic Runge-Kutta (SRK2) to approximate the solution of stochastic delay differential equations (SDDEs) with a constant time lag, r > 0. General formulation of stochastic Runge-Kutta for SDDEs is introduced and Stratonovich Taylor series expansion for numerical solution of SRK2 is presented. Local truncation error of SRK2 is measured by comparing the Stratonovich Taylor expansion of the exact solution with the computed solution. Numerical experiment is performed to assure the validity of the method in simulating the strong solution of SDDEs.

Using delay differential equations to induce alternans in a model of cardiac electrophysiology.

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Eastman, Justin; Sass, Julian; Gomes, Johnny M; Dos Santos, Rodrigo Weber; Cherry, Elizabeth M

2016-09-07

Cardiac electrical alternans is a period-2 dynamical behavior with alternating long and short action potential durations (APD) that often precedes dangerous arrhythmias associated with cardiac arrest. Despite the importance of alternans, many current ordinary differential equations models of cardiac electrophysiology do not produce alternans, thereby limiting the use of these models for studying the mechanisms that underlie this condition. Because delay differential equations (DDEs) commonly induce complex dynamics in other biological systems, we investigate whether incorporating DDEs can lead to alternans development in cardiac models by studying the Fox et al. canine ventricular action potential model. After suppressing the alternans in the original model, we show that alternans can be obtained by introducing DDEs in the model gating variables, and we quantitatively compare the DDE-induced alternans with the alternans present in the original model. We analyze the behavior of the voltage, currents, and gating variables of the model to study the effects of the delays and to determine how alternans develops in that setting, and we discuss the mathematical and physiological implications of our findings. In future work, we aim to apply our approach to induce alternans in models that do not naturally exhibit such dynamics. Copyright © 2016 Elsevier Ltd. All rights reserved.

Antiperiodic Solutions for a Kind of Nonlinear Duffing Equations with a Deviating Argument and Time-Varying Delay

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Changjin Xu

2014-01-01

Full Text Available This paper deals with a kind of nonlinear Duffing equation with a deviating argument and time-varying delay. By using differential inequality techniques, some very verifiable criteria on the existence and exponential stability of antiperiodic solutions for the equation are obtained. Our results are new and complementary to previously known results. An example is given to illustrate the feasibility and effectiveness of our main results.

About Positivity of Green's Functions for Nonlocal Boundary Value Problems with Impulsive Delay Equations

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Alexander Domoshnitsky

2014-01-01

Full Text Available The impulsive delay differential equation is considered (Lx(t=x′(t+∑i=1mpi(tx(t-τi(t=f(t, t∈[a,b],  x(tj=βjx(tj-0, j=1,…,k, a=t0equation are obtained.

Chenciner bubbles and torus break-up in a periodically forced delay differential equation

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Keane, A.; Krauskopf, B.

2018-06-01

We study a generic model for the interaction of negative delayed feedback and periodic forcing that was first introduced by Ghil et al (2008 Nonlinear Process. Geophys. 15 417–33) in the context of the El Niño Southern Oscillation climate system. This model takes the form of a delay differential equation and has been shown in previous work to be capable of producing complicated dynamics, which is organised by resonances between the external forcing and dynamics induced by feedback. For certain parameter values, we observe in simulations the sudden disappearance of (two-frequency dynamics on) tori. This can be explained by the folding of invariant tori and their associated resonance tongues. It is known that two smooth tori cannot simply meet and merge; they must actually break up in complicated bifurcation scenarios that are organised within so-called resonance bubbles first studied by Chenciner. We identify and analyse such a Chenciner bubble in order to understand the dynamics at folds of tori. We conduct a bifurcation analysis of the Chenciner bubble by means of continuation software and dedicated simulations, whereby some bifurcations involve tori and are detected in appropriate two-dimensional projections associated with Poincaré sections. We find close agreement between the observed bifurcation structure in the Chenciner bubble and a previously suggested theoretical picture. As far as we are aware, this is the first time the bifurcation structure associated with a Chenciner bubble has been analysed in a delay differential equation and, in fact, for a flow rather than an explicit map. Following our analysis, we briefly discuss the possible role of folding tori and Chenciner bubbles in the context of tipping.

Bifurcation analysis of a neutral delay differential equation modelling the torsional motion of a driven drill-string

Energy Technology Data Exchange (ETDEWEB)

Balanov, A.G.; Janson, N.B. E-mail: n.janson@lancaster.ac.uk; McClintock, P.V.E.; Tucker, R.W.; Wang, C.H.T

2003-01-01

Using techniques from dynamical systems analysis we explore numerically the solution space, under parametric variation, of a neutral differential delay equation that arises naturally in the Cosserat description of torsional waves on a driven drill-string.

Bifurcation analysis of a neutral delay differential equation modelling the torsional motion of a driven drill-string

International Nuclear Information System (INIS)

Balanov, A.G.; Janson, N.B.; McClintock, P.V.E.; Tucker, R.W.; Wang, C.H.T.

2003-01-01

Using techniques from dynamical systems analysis we explore numerically the solution space, under parametric variation, of a neutral differential delay equation that arises naturally in the Cosserat description of torsional waves on a driven drill-string

Existence of mild solutions for nonlocal Cauchy problem for fractional neutral evolution equations with infinite delay

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V. Vijayakumar

2014-09-01

Full Text Available In this article, we study the existence of mild solutions for nonlocal Cauchy problem for fractional neutral evolution equations with infinite delay. The results are obtained by using the Banach contraction principle. Finally, an application is given to illustrate the theory.

Well-posedness and exponential stability for a wave equation with nonlocal time-delay condition

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Carlos Alberto Raposo

2017-11-01

Full Text Available Well-posedness and exponential stability of nonlocal time-delayed of a wave equation with a integral conditions of the 1st kind forms the center of this work. Through semigroup theory we prove the well-posedness by the Hille-Yosida theorem and the exponential stability exploring the dissipative properties of the linear operator associated to damped model using the Gearhart-Huang-Pruss theorem.

Periodic solutions for a non-monotone family of delayed differential equations with applications to Nicholson systems

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Faria, Teresa

2017-07-01

For a family of n-dimensional periodic delay differential equations which encompasses a broad set of models used in structured population dynamics, the existence of a positive periodic solution is obtained under very mild conditions. The proof uses the Schauder fixed point theorem and relies on the permanence of the system. A general criterion for the existence of a positive periodic solution for Nicholson's blowflies periodic systems (with both distributed and discrete time-varying delays) is derived as a simple application of our main result, generalizing the few existing results concerning multi-dimensional Nicholson models. In the case of a Nicholson system with discrete delays all multiples of the period, the global attractivity of the positive periodic solution is further analyzed, improving results in recent literature.

A numerical solution for a class of time fractional diffusion equations with delay

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Pimenov Vladimir G.

2017-09-01

Full Text Available This paper describes a numerical scheme for a class of fractional diffusion equations with fixed time delay. The study focuses on the uniqueness, convergence and stability of the resulting numerical solution by means of the discrete energy method. The derivation of a linearized difference scheme with convergence order O(τ2−α+ h4 in L∞-norm is the main purpose of this study. Numerical experiments are carried out to support the obtained theoretical results.

Stability and square integrability of derivatives of solutions of nonlinear fourth order differential equations with delay.

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Korkmaz, Erdal

2017-01-01

In this paper, we give sufficient conditions for the boundedness, uniform asymptotic stability and square integrability of the solutions to a certain fourth order non-autonomous differential equations with delay by using Lyapunov's second method. The results obtained essentially improve, include and complement the results in the literature.

Stability and bifurcation of numerical discretization of a second-order delay differential equation with negative feedback

International Nuclear Information System (INIS)

Ding Xiaohua; Su Huan; Liu Mingzhu

2008-01-01

The paper analyzes a discrete second-order, nonlinear delay differential equation with negative feedback. The characteristic equation of linear stability is solved, as a function of two parameters describing the strength of the feedback and the damping in the autonomous system. The existence of local Hopf bifurcations is investigated, and the direction and stability of periodic solutions bifurcating from the Hopf bifurcation of the discrete model are determined by the Hopf bifurcation theory of discrete system. Finally, some numerical simulations are performed to illustrate the analytical results found

Controllability for Semilinear Functional and Neutral Functional Evolution Equations with Infinite Delay in Frechet Spaces

International Nuclear Information System (INIS)

Agarwal, Ravi P.; Baghli, Selma; Benchohra, Mouffak

2009-01-01

The controllability of mild solutions defined on the semi-infinite positive real interval for two classes of first order semilinear functional and neutral functional differential evolution equations with infinite delay is studied in this paper. Our results are obtained using a recent nonlinear alternative due to Avramescu for sum of compact and contraction operators in Frechet spaces, combined with the semigroup theory

Stability and square integrability of derivatives of solutions of nonlinear fourth order differential equations with delay

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Erdal Korkmaz

2017-06-01

Full Text Available Abstract In this paper, we give sufficient conditions for the boundedness, uniform asymptotic stability and square integrability of the solutions to a certain fourth order non-autonomous differential equations with delay by using Lyapunov’s second method. The results obtained essentially improve, include and complement the results in the literature.

Delay equations modeling the effects of phase-specific drugs and immunotherapy on proliferating tumor cells.

Science.gov (United States)

Barbarossa, Maria Vittoria; Kuttler, Christina; Zinsl, Jonathan

2012-04-01

In this work we present a mathematical model for tumor growth based on the biology of the cell cycle. For an appropriate description of the effects of phase-specific drugs, it is necessary to look at the cell cycle and its phases. Our model reproduces the dynamics of three different tumor cell populations: quiescent cells, cells during the interphase and mitotic cells. Starting from a partial differential equations (PDEs) setting, a delay differential equations (DDE) model is derived for an easier and more realistic approach. Our equations also include interactions of tumor cells with immune system effectors. We investigate the model both from the analytical and the numerical point of view, give conditions for positivity of solutions and focus on the stability of the cancer-free equilibrium. Different immunotherapeutic strategies and their effects on the tumor growth are considered, as well.

The existence and global exponential stability of a periodic solution of a class of delay differential equations

International Nuclear Information System (INIS)

Tang, X H; Zou, Xingfu

2009-01-01

By employing Schauder's fixed point theorem and a non-Liapunov method (matrix theory, inequality analysis), we obtain some new criteria that ensure existence and global exponential stability of a periodic solution to a class of functional differential equations. Applying these criteria to a cellular neural network with time delays (delayed cellular neural network, DCNN) under a periodic environment leads to some new results that improve and generalize many existing ones we know on this topic. These results are of great significance in designs and applications of globally stable periodic DCNNs

Asymptotic Comparison of the Solutions of Linear Time-Delay Systems with Point and Distributed Lags with Those of Their Limiting Equations

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M. De la Sen

2009-01-01

Full Text Available This paper investigates the relations between the particular eigensolutions of a limiting functional differential equation of any order, which is the nominal (unperturbed linear autonomous differential equations, and the associate ones of the corresponding perturbed functional differential equation. Both differential equations involve point and distributed delayed dynamics including Volterra class dynamics. The proofs are based on a Perron-type theorem for functional equations so that the comparison is governed by the real part of a dominant zero of the characteristic equation of the nominal differential equation. The obtained results are also applied to investigate the global stability of the perturbed equation based on that of its corresponding limiting equation.

T-Stability of the Heun Method and Balanced Method for Solving Stochastic Differential Delay Equations

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Xiaolin Zhu

2014-01-01

Full Text Available This paper studies the T-stability of the Heun method and balanced method for solving stochastic differential delay equations (SDDEs. Two T-stable conditions of the Heun method are obtained for two kinds of linear SDDEs. Moreover, two conditions under which the balanced method is T-stable are obtained for two kinds of linear SDDEs. Some numerical examples verify the theoretical results proposed.

Multifractal chaotic attractors in a system of delay-differential equations modeling road traffic.

Science.gov (United States)

Safonov, Leonid A.; Tomer, Elad; Strygin, Vadim V.; Ashkenazy, Yosef; Havlin, Shlomo

2002-12-01

We study a system of delay-differential equations modeling single-lane road traffic. The cars move in a closed circuit and the system's variables are each car's velocity and the distance to the car ahead. For low and high values of traffic density the system has a stable equilibrium solution, corresponding to the uniform flow. Gradually decreasing the density from high to intermediate values we observe a sequence of supercritical Hopf bifurcations forming multistable limit cycles, corresponding to flow regimes with periodically moving traffic jams. Using an asymptotic technique we find approximately small limit cycles born at Hopf bifurcations and numerically preform their global continuations with decreasing density. For sufficiently large delay the system passes to chaos following the Ruelle-Takens-Newhouse scenario (limit cycles-two-tori-three-tori-chaotic attractors). We find that chaotic and nonchaotic attractors coexist for the same parameter values and that chaotic attractors have a broad multifractal spectrum. (c) 2002 American Institute of Physics.

A new multi-step technique with differential transform method for analytical solution of some nonlinear variable delay differential equations.

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Benhammouda, Brahim; Vazquez-Leal, Hector

2016-01-01

This work presents an analytical solution of some nonlinear delay differential equations (DDEs) with variable delays. Such DDEs are difficult to treat numerically and cannot be solved by existing general purpose codes. A new method of steps combined with the differential transform method (DTM) is proposed as a powerful tool to solve these DDEs. This method reduces the DDEs to ordinary differential equations that are then solved by the DTM. Furthermore, we show that the solutions can be improved by Laplace-Padé resummation method. Two examples are presented to show the efficiency of the proposed technique. The main advantage of this technique is that it possesses a simple procedure based on a few straight forward steps and can be combined with any analytical method, other than the DTM, like the homotopy perturbation method.

Exponential Stability of the Monotubular Heat Exchanger Equation with Time Delay in Boundary Observation

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Xue-Lian Jin

2017-01-01

Full Text Available The exponential stability of the monotubular heat exchanger equation with boundary observation possessing a time delay and inner control was investigated. Firstly, the close-loop system was translated into an abstract Cauchy problem in the suitable state space. A uniformly bounded C0-semigroup generated by the close-loop system, which implies that the unique solution of the system exists, was shown. Secondly, the spectrum configuration of the closed-loop system was analyzed and the eventual differentiability and the eventual compactness of the semigroup were shown by the resolvent estimates on some resolvent sets. This implies that the spectrum-determined growth assumption holds. Finally, a sufficient condition, which is related to the physical parameters in the system and is independent of the time delay, of the exponential stability of the closed-loop system was given.

Numerical Oscillations Analysis for Nonlinear Delay Differential Equations in Physiological Control Systems

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Qi Wang

2012-01-01

Full Text Available This paper deals with the oscillations of numerical solutions for the nonlinear delay differential equations in physiological control systems. The exponential θ-method is applied to p′(t=β0ωμp(t−τ/(ωμ+pμ(t−τ−γp(t and it is shown that the exponential θ-method has the same order of convergence as that of the classical θ-method. Several conditions under which the numerical solutions oscillate are derived. Moreover, it is proven that every nonoscillatory numerical solution tends to positive equilibrium of the continuous system. Finally, the main results are illustrated with numerical examples.

Application of the variational iteration method for system of initial value problems delay differential equations

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Yousef, Hamood. M.; Ismail, A. I. B. MD.

2017-08-01

Many attempts have been presented to solve the system of Delay Differential Equations (DDE) with Initial Value Problem. As a result, it has shown difficulties when getting the solution or cannot be solved. In this paper, a Variational Iteration Method is employed to find out an approximate solution for the system of DDE with initial value problems. The example illustrates convenient and an efficiency comparison with the exact solution.

Time-delay equation governing electron motion

International Nuclear Information System (INIS)

Cohn, J.

1976-01-01

A previously proposed differential-difference equation governing the motion of the classical radiating electron is considered further. A set of three assumptions is offered, under which the proposed equation yields asymptotically stable acceleration

A Four-Stage Fifth-Order Trigonometrically Fitted Semi-Implicit Hybrid Method for Solving Second-Order Delay Differential Equations

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Sufia Zulfa Ahmad

2016-01-01

Full Text Available We derived a two-step, four-stage, and fifth-order semi-implicit hybrid method which can be used for solving special second-order ordinary differential equations. The method is then trigonometrically fitted so that it is suitable for solving problems which are oscillatory in nature. The methods are then used for solving oscillatory delay differential equations. Numerical results clearly show the efficiency of the new method when compared to the existing explicit and implicit methods in the scientific literature.

Fitted Fourier-pseudospectral methods for solving a delayed reaction-diffusion partial differential equation in biology

Science.gov (United States)

Adam, A. M. A.; Bashier, E. B. M.; Hashim, M. H. A.; Patidar, K. C.

2017-07-01

In this work, we design and analyze a fitted numerical method to solve a reaction-diffusion model with time delay, namely, a delayed version of a population model which is an extension of the logistic growth (LG) equation for a food-limited population proposed by Smith [F.E. Smith, Population dynamics in Daphnia magna and a new model for population growth, Ecology 44 (1963) 651-663]. Seeing that the analytical solution (in closed form) is hard to obtain, we seek for a robust numerical method. The method consists of a Fourier-pseudospectral semi-discretization in space and a fitted operator implicit-explicit scheme in temporal direction. The proposed method is analyzed for convergence and we found that it is unconditionally stable. Illustrative numerical results will be presented at the conference.

Interval Oscillation Criteria for Super-Half-Linear Impulsive Differential Equations with Delay

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Zhonghai Guo

2012-01-01

Full Text Available We study the following second-order super-half-linear impulsive differential equations with delay [r(tφγ(x′(t]′+p(tφγ(x(t-σ+q(tf(x(t-σ=e(t, t≠τk, x(t+=akx(t, x′(t+=bkx′(t, t=τk, where t≥t0∈ℝ, φ*(u=|u|*-1u, σ is a nonnegative constant, {τk} denotes the impulsive moments sequence with τ1σ. By some classical inequalities, Riccati transformation, and two classes of functions, we give several interval oscillation criteria which generalize and improve some known results. Moreover, we also give two examples to illustrate the effectiveness and nonemptiness of our results.

Adaptive modification of the delayed feedback control algorithm with a continuously varying time delay

International Nuclear Information System (INIS)

Pyragas, V.; Pyragas, K.

2011-01-01

We propose a simple adaptive delayed feedback control algorithm for stabilization of unstable periodic orbits with unknown periods. The state dependent time delay is varied continuously towards the period of controlled orbit according to a gradient-descent method realized through three simple ordinary differential equations. We demonstrate the efficiency of the algorithm with the Roessler and Mackey-Glass chaotic systems. The stability of the controlled orbits is proven by computation of the Lyapunov exponents of linearized equations. -- Highlights: → A simple adaptive modification of the delayed feedback control algorithm is proposed. → It enables the control of unstable periodic orbits with unknown periods. → The delay time is varied continuously according to a gradient descend method. → The algorithm is embodied by three simple ordinary differential equations. → The validity of the algorithm is proven by computation of the Lyapunov exponents.

Delayed Stochastic Linear-Quadratic Control Problem and Related Applications

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Li Chen

2012-01-01

stochastic differential equations (FBSDEs with Itô’s stochastic delay equations as forward equations and anticipated backward stochastic differential equations as backward equations. Especially, we present the optimal feedback regulator for the time delay system via a new type of Riccati equations and also apply to a population optimal control problem.

Development of kinetics equations from the Boltzmann equation; Etablissement des equations de la cinetique a partir de l'equation de Boltzmann

Energy Technology Data Exchange (ETDEWEB)

Plas, R.

1962-07-01

The author reports a study on kinetics equations for a reactor. He uses the conventional form of these equations but by using a dynamic multiplication factor. Thus, constants related to delayed neutrons are not modified by efficiency factors. The author first describes the theoretic kinetic operation of a reactor and develops the associated equations. He reports the development of equations for multiplication factors.

Almost-Periodic Weak Solutions of Second-Order Neutral Delay-Differential Equations with Piecewise Constant Argument

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Chuanyi Zhang

2008-06-01

Full Text Available We investigate the existence of almost-periodic weak solutions of second-order neutral delay-differential equations with piecewise constant argument of the form (x(t+x(t−1′′=qx(2[(t+1/2]+f(t, where [⋅] denotes the greatest integer function, q is a real nonzero constant, and f(t is almost periodic.

The application of Legendre-tau approximation to parameter identification for delay and partial differential equations

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Ito, K.

1983-01-01

Approximation schemes based on Legendre-tau approximation are developed for application to parameter identification problem for delay and partial differential equations. The tau method is based on representing the approximate solution as a truncated series of orthonormal functions. The characteristic feature of the Legendre-tau approach is that when the solution to a problem is infinitely differentiable, the rate of convergence is faster than any finite power of 1/N; higher accuracy is thus achieved, making the approach suitable for small N.

A Note on a Semilinear Fractional Differential Equation of Neutral Type with Infinite Delay

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Gisle M. Mophou

2010-01-01

Full Text Available We deal in this paper with the mild solution for the semilinear fractional differential equation of neutral type with infinite delay: Dαx(t+Ax(t=f(t,xt, t∈[0,T], x(t=ϕ(t, t∈]−∞,0], with T>0 and 0<α<1. We prove the existence (and uniqueness of solutions, assuming that −A is a linear closed operator which generates an analytic semigroup (T(tt≥0 on a Banach space 𝕏 by means of the Banach's fixed point theorem. This generalizes some recent results.

Positive solutions of a three-point boundary-value problem for differential equations with damping and actively bounded delayed forcing term

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George L. Karakostas

2006-08-01

Full Text Available We provide sufficient conditions for the existence of positive solutions of a three-point boundary value problem concerning a second order delay differential equation with damping and forcing term whose the delayed part is an actively bounded function, a meaning introduced in [19]. By writing the damping term as a difference of two factors one can extract more information on the solutions. (For instance, in an application, given in the last section, we can give the exact value of the norm of the solution.

Testing electric field models using ring current ion energy spectra from the Equator-S ion composition (ESIC instrument

Directory of Open Access Journals (Sweden)

L. M. Kistler

Full Text Available During the main and early recovery phase of a geomagnetic storm on February 18, 1998, the Equator-S ion composition instrument (ESIC observed spectral features which typically represent the differences in loss along the drift path in the energy range (5–15 keV/e where the drift changes from being E × B dominated to being gradient and curvature drift dominated. We compare the expected energy spectra modeled using a Volland-Stern electric field and a Weimer electric field, assuming charge exchange along the drift path, with the observed energy spectra for H⁺ and O⁺. We find that using the Weimer electric field gives much better agreement with the spectral features, and with the observed losses. Neither model, however, accurately predicts the energies of the observed minima.

Key words. Magnetospheric physics (energetic particles trapped; plasma convection; storms and substorms

The magnetic field experiment onboard Equator-S and its scientific possibilities

Directory of Open Access Journals (Sweden)

K.-H. Fornacon

1999-12-01

Full Text Available The special feature of the ringcore fluxgate magnetometer on Equator-S is the high time and field resolution. The scientific aim of the experiment is the investigation of waves in the 10–100 picotesla range with a time resolution up to 64 Hz. The instrument characteristics and the influence of the spacecraft on the magnetic field measurement will be discussed. The work shows that the applied pre- and inflight calibration techniques are sufficient to suppress spacecraft interferences. The offset in spin axis direction was determined for the first time with an independent field measurement by the Equator-S Electron Drift Instrument. The data presented gives an impression of the accuracy of the measurement.Key words. Magnetospheric physics (instruments and techniques · Space plasma physics (instruments and techniques

The magnetic field experiment onboard Equator-S and its scientific possibilities

Directory of Open Access Journals (Sweden)

K.-H. Fornacon

Full Text Available The special feature of the ringcore fluxgate magnetometer on Equator-S is the high time and field resolution. The scientific aim of the experiment is the investigation of waves in the 10–100 picotesla range with a time resolution up to 64 Hz. The instrument characteristics and the influence of the spacecraft on the magnetic field measurement will be discussed. The work shows that the applied pre- and inflight calibration techniques are sufficient to suppress spacecraft interferences. The offset in spin axis direction was determined for the first time with an independent field measurement by the Equator-S Electron Drift Instrument. The data presented gives an impression of the accuracy of the measurement.

Key words. Magnetospheric physics (instruments and techniques · Space plasma physics (instruments and techniques

The role of delay in the dynamics of nuclear reactors

International Nuclear Information System (INIS)

Svitra, D.; Bucys, K.

1999-01-01

The stability of nuclear reactors based on nonlinear models of reactor dynamics including the action of delayed neutrons is analysed. The point model of reactor dynamics with the system of seven nonlinear simple differential equations was changed to the system of two nonlinear differential equations including the action of delay. The method of the theory of bifurcations for nonlinear differential equations with delay is used. (author)

Communication key using delay times in time-delayed chaos synchronization

International Nuclear Information System (INIS)

Kim, Chil-Min; Kye, Won-Ho; Rim, Sunghwan; Lee, Soo-Young

2004-01-01

We propose an efficient key scheme, which can generate a great number of communication keys, for communication using chaos synchronization. We have attained the keys from delay times of time-delay coupled chaotic systems. We explain the scheme and the efficiency by coupling Henon and logistic maps and illustrate them by coupling Navier-Stokes and Lorenz equations as a continuous system

COMPARISON THEOREMS AND APPLICATIONS OF OSCILLATION OF NEUTRAL DIFFERENTIAL EQUATIONS

Institute of Scientific and Technical Information of China (English)

燕居让

1991-01-01

We first establish comparison theorems of the oscillation for a higher-order neutral delaydifferential equation. By these comparison theorems, the criterion of oscillation propertiesof neutral delay differential equation is reduced to that of nonneutral delay differential equa-tion, from which we give a series of oscillation theorems for neutral delay differentialequation.

First-order differential-delay equation for the baroreflex predicts the 0.4-Hz blood pressure rhythm in rats.

Science.gov (United States)

Burgess, D E; Hundley, J C; Li, S G; Randall, D C; Brown, D R

1997-12-01

We have described a 0.4-Hz rhythm in renal sympathetic nerve activity (SNA) that is tightly coupled to 0.4-Hz oscillations in blood pressure in the unanesthetized rat. In previous work, the relationship between SNA and fluctuations in mean arterial blood pressure (MAP) was described by a set of two first-order differential equations. We have now modified our earlier model to test the feasibility that the 0.4-Hz rhythm can be explained by the baroreflex without requiring a neural oscillator. In this baroreflex model, a linear feedback term replaces the sympathetic drive to the cardiovascular system. The time delay in the feedback loop is set equal to the time delay on the efferent side, approximately 0.5 s (as determined in the initial model), plus a time delay of 0.2 s on the afferent side for a total time delay of approximately 0.7 s. A stability analysis of this new model yields feedback resonant frequencies close to 0.4 Hz. Because of the time delay in the feedback loop, the proportional gain may not exceed a value on the order of 10 to maintain stability. The addition of a derivative feedback term increases the system's stability for a positive range of derivative gains. We conclude that the known physiological time delay for the sympathetic portion of the baroreflex can account for the observed 0.4-Hz rhythm in rat MAP and that the sensitivity of the baroreceptors to the rate of change in blood pressure, as well as average blood pressure, would enhance the natural stability of the baroreflex.

A stability criterion for HNFDE with non-uniform delays

International Nuclear Information System (INIS)

Liu Xingwen; Zhong Shouming; Zhang Fengli

2005-01-01

Stability of functional differential equations (FDE) is an increasingly important problem in both science and engineering. Delays, whether uniform or non-uniform, play an important role in the dynamics of a system. Since non-uniform delay is more general and less focused than uniform delay, this paper concentrates on the stability of high-order neutral functional differential equations (HNFDE) with non-uniform delay, and proposes a sufficient condition for it. This result may be widely helpful, thanks to the frequent emergence of a HNFDE with non-uniform delay in various fields. Its effectiveness is illustrated by some examples

Application of a Lie group admitted by a homogeneous equation for group classification of a corresponding inhomogeneous equation

Science.gov (United States)

Long, Feng-Shan; Karnbanjong, Adisak; Suriyawichitseranee, Amornrat; Grigoriev, Yurii N.; Meleshko, Sergey V.

2017-07-01

This paper proposes an algorithm for group classification of a nonhomogeneous equation using the group analysis provided for the corresponding homogeneous equation. The approach is illustrated by a partial differential equation, an integro-differential equation, and a delay partial differential equation.

Periodic solutions in reaction–diffusion equations with time delay

International Nuclear Information System (INIS)

Li, Li

2015-01-01

Spatial diffusion and time delay are two main factors in biological and chemical systems. However, the combined effects of them on diffusion systems are not well studied. As a result, we investigate a nonlinear diffusion system with delay and obtain the existence of the periodic solutions using coincidence degree theory. Moreover, two numerical examples confirm our theoretical results. The obtained results can also be applied in other related fields

Positive Periodic Solution for the Generalized Neutral Differential Equation with Multiple Delays and Impulse

Directory of Open Access Journals (Sweden)

Zhenguo Luo

2014-01-01

Full Text Available By using a fixed point theorem of strict-set-contraction, which is different from Gaines and Mawhin's continuation theorem and abstract continuation theory for k-set contraction, we established some new criteria for the existence of positive periodic solution of the following generalized neutral delay functional differential equation with impulse: x'(t=x(t[a(t-f(t,x(t,x(t-τ1(t,x(t,…,x(t-τn(t,x(t,x'(t-γ1(t,x(t,…,x'(t-γm(t,x(t],  t≠tk,  k∈Z+;  x(tk+=x(tk-+θk(x(tk,  k∈Z+. As applications of our results, we also give some applications to several Lotka-Volterra models and new results are obtained.

Linear measure functional differential equations with infinite delay

Czech Academy of Sciences Publication Activity Database

Monteiro, Giselle Antunes; Slavík, A.

2014-01-01

Roč. 287, 11-12 (2014), s. 1363-1382 ISSN 0025-584X Institutional support: RVO:67985840 Keywords : measure functional differential equations * generalized ordinary differential equations * Kurzweil-Stieltjes integral Subject RIV: BA - General Mathematics Impact factor: 0.683, year: 2014 http://onlinelibrary.wiley.com/doi/10.1002/mana.201300048/abstract

Time-Delay Effects on Constitutive Gene Expression*

International Nuclear Information System (INIS)

Feng Yan-Ling; Wang Dan; Tang Xu-Lei; Dong Jian-Min

2017-01-01

The dynamics of constitutive gene expression with delayed mRNA degradation is investigated, where the intrinsic noise caused by the small number of reactant molecules is introduced. It is found that the oscillatory behavior claimed in previous investigations does not appear in the approximation of small time delay, and the steady state distribution still follows the Poisson law. Furthermore, we introduce the extrinsic noise induced by surrounding environment to explore the effects of this noise and time delay on the Fano factor. Based on a delay Langevin equation and the corresponding Fokker–Planck equation, the distribution of mRNA copy-number is achieved analytically. The time delay and extrinsic noise play similar roles in the gene expression system, that is, they are able to result in the deviation of the Fano factor from 1 evidently. The measured Fano factor for constitutive gene expression is slightly larger than 1, which is perhaps attributed to the time-delay effect. (paper)

Adding delayed recall to the ADAS-cog improves measurement precision in mild Alzheimer's disease: Implications for predicting instrumental activities of daily living.

Science.gov (United States)

Lowe, Deborah A; Balsis, Steve; Benge, Jared F; Doody, Rachelle S

2015-12-01

As research increasingly focuses on preclinical stages of Alzheimer's disease (AD), instruments must be retooled to identify early cognitive markers of AD. A supplemental delayed recall subtest for the Alzheimer's Disease Assessment Scale-cognitive (ADAS-cog; Mohs, Rosen, & Davis, 1983; Rosen, Mohs, & Davis, 1984) is commonly implemented, but it is not known precisely where along the spectrum of cognitive dysfunction this subtest yields incremental information beyond what is gained from the standard ADAS-cog, or whether it can improve prediction of functional outcomes. An item response theory approach can analyze this in a psychometrically rigorous way. Seven hundred eighty-eight patients with AD or amnestic complaints or impairment completed a battery including the ADAS-cog and 2 activities of daily living measures. The delayed recall subtest slightly improved the ADAS-cog's measurement precision in the mild range of cognitive dysfunction and increased prediction of instrumental activities of daily living for individuals with subjective memory impairment. (c) 2015 APA, all rights reserved).

Functional equations with causal operators

CERN Document Server

Corduneanu, C

2003-01-01

Functional equations encompass most of the equations used in applied science and engineering: ordinary differential equations, integral equations of the Volterra type, equations with delayed argument, and integro-differential equations of the Volterra type. The basic theory of functional equations includes functional differential equations with causal operators. Functional Equations with Causal Operators explains the connection between equations with causal operators and the classical types of functional equations encountered by mathematicians and engineers. It details the fundamentals of linear equations and stability theory and provides several applications and examples.

Finite-Time Attractivity for Diagonally Dominant Systems with Off-Diagonal Delays

Directory of Open Access Journals (Sweden)

T. S. Doan

2012-01-01

Full Text Available We introduce a notion of attractivity for delay equations which are defined on bounded time intervals. Our main result shows that linear delay equations are finite-time attractive, provided that the delay is only in the coupling terms between different components, and the system is diagonally dominant. We apply this result to a nonlinear Lotka-Volterra system and show that the delay is harmless and does not destroy finite-time attractivity.

Exponential synchronization of two nonlinearly non-delayed and delayed coupled complex dynamical networks

International Nuclear Information System (INIS)

Zheng Song

2012-01-01

In this paper, the exponential synchronization between two nonlinearly coupled complex networks with non-delayed and delayed coupling is investigated with Lyapunov-Krasovskii-type functionals. Based on the stability analysis of the impulsive differential equation and the linear matrix inequality, sufficient delay-dependent conditions for exponential synchronization are derived, and a linear impulsive controller and simple updated laws are also designed. Furthermore, the coupling matrices need not be symmetric or irreducible. Numerical examples are presented to verify the effectiveness and correctness of the synchronization criteria obtained.

Bifurcation analysis of delay-induced periodic oscillations

NARCIS (Netherlands)

Green, K.

2010-01-01

In this paper we consider a generic differential equation with a cubic nonlinearity and delay. This system, in the absence of delay, is known to undergo an oscillatory instability. The addition of the delay is shown to result in the creation of a number of periodic solutions with constant amplitude

Lag synchronization of chaotic systems with time-delayed linear ...

Indian Academy of Sciences (India)

delayed linear terms via impulsive control is investigated. Based on the stability theory of impulsive delayed differen- tial equations, some sufficient conditions are obtained guaranteeing the synchronized behaviours between two delayed chaotic ...

Kinetics of intravenous radiographic contrast medium injections as used on CT: simulation with time delay differential equations in a basic human cardiovascular multicompartment model.

Science.gov (United States)

Violon, D

2012-12-01

To develop a multicompartment model of only essential human body components that predicts the contrast medium concentration vs time curve in a chosen compartment after an intravenous injection. Also to show that the model can be used to time adequately contrast-enhanced CT series. A system of linked time delay instead of ordinary differential equations described the model and was solved with a Matlab program (Matlab v. 6.5; The Mathworks, Inc., Natick, MA). All the injection and physiological parameters were modified to cope with normal or pathological situations. In vivo time-concentration curves from the literature were recalculated to validate the model. The recalculated contrast medium time-concentration curves and parameters are given. The results of the statistical analysis of the study findings are expressed as the median prediction error and the median absolute prediction error values for both the time delay and ordinary differential equation systems; these are situated well below the generally accepted maximum 20% limit. The presented program correctly predicts the time-concentration curve of an intravenous contrast medium injection and, consequently, allows an individually tailored approach of CT examinations with optimised use of the injected contrast medium volume, as long as time delay instead of ordinary differential equations are used. The presented program offers good preliminary knowledge of the time-contrast medium concentration curve after any intravenous injection, allowing adequate timing of a CT examination, required by the short scan time of present-day scanners. The injected volume of contrast medium can be tailored to the individual patient with no more contrast medium than is strictly needed.

Approximate Solutions of Delay Differential Equations with Constant and Variable Coefficients by the Enhanced Multistage Homotopy Perturbation Method

Directory of Open Access Journals (Sweden)

D. Olvera

2015-01-01

Full Text Available We expand the application of the enhanced multistage homotopy perturbation method (EMHPM to solve delay differential equations (DDEs with constant and variable coefficients. This EMHPM is based on a sequence of subintervals that provide approximate solutions that require less CPU time than those computed from the dde23 MATLAB numerical integration algorithm solutions. To address the accuracy of our proposed approach, we examine the solutions of several DDEs having constant and variable coefficients, finding predictions with a good match relative to the corresponding numerical integration solutions.

Quasi-Newton methods for parameter estimation in functional differential equations

Science.gov (United States)

Brewer, Dennis W.

1988-01-01

A state-space approach to parameter estimation in linear functional differential equations is developed using the theory of linear evolution equations. A locally convergent quasi-Newton type algorithm is applied to distributed systems with particular emphasis on parameters that induce unbounded perturbations of the state. The algorithm is computationally implemented on several functional differential equations, including coefficient and delay estimation in linear delay-differential equations.

Numerical Solution of Piecewise Constant Delay Systems Based on a Hybrid Framework

Directory of Open Access Journals (Sweden)

H. R. Marzban

2016-01-01

Full Text Available An efficient numerical scheme for solving delay differential equations with a piecewise constant delay function is developed in this paper. The proposed approach is based on a hybrid of block-pulse functions and Taylor’s polynomials. The operational matrix of delay corresponding to the proposed hybrid functions is introduced. The sparsity of this matrix significantly reduces the computation time and memory requirement. The operational matrices of integration, delay, and product are employed to transform the problem under consideration into a system of algebraic equations. It is shown that the developed approach is also applicable to a special class of nonlinear piecewise constant delay differential equations. Several numerical experiments are examined to verify the validity and applicability of the presented technique.

Adjoint method provides phase response functions for delay-induced oscillations.

Science.gov (United States)

Kotani, Kiyoshi; Yamaguchi, Ikuhiro; Ogawa, Yutaro; Jimbo, Yasuhiko; Nakao, Hiroya; Ermentrout, G Bard

2012-07-27

Limit-cycle oscillations induced by time delay are widely observed in various systems, but a systematic phase-reduction theory for them has yet to be developed. Here we present a practical theoretical framework to calculate the phase response function Z(θ), a fundamental quantity for the theory, of delay-induced limit cycles with infinite-dimensional phase space. We show that Z(θ) can be obtained as a zero eigenfunction of the adjoint equation associated with an appropriate bilinear form for the delay differential equations. We confirm the validity of the proposed framework for two biological oscillators and demonstrate that the derived phase equation predicts intriguing multimodal locking behavior.

Modeling endocrine regulation of the menstrual cycle using delay differential equations.

Science.gov (United States)

Harris, Leona A; Selgrade, James F

2014-11-01

This article reviews an effective mathematical procedure for modeling hormonal regulation of the menstrual cycle of adult women. The procedure captures the effects of hormones secreted by several glands over multiple time scales. The specific model described here consists of 13 nonlinear, delay, differential equations with 44 parameters and correctly predicts blood levels of ovarian and pituitary hormones found in the biological literature for normally cycling women. In addition to this normal cycle, the model exhibits another stable cycle which may describe a biologically feasible "abnormal" condition such as polycystic ovarian syndrome. Model simulations illustrate how one cycle can be perturbed to the other cycle. Perturbations due to the exogenous administration of each ovarian hormone are examined. This model may be used to test the effects of hormone therapies on abnormally cycling women as well as the effects of exogenous compounds on normally cycling women. Sensitive parameters are identified and bifurcations in model behavior with respect to parameter changes are discussed. Modeling various aspects of menstrual cycle regulation should be helpful in predicting successful hormone therapies, in studying the phenomenon of cycle synchronization and in understanding many factors affecting the aging of the female reproductive endocrine system. Copyright © 2014 Elsevier Inc. All rights reserved.

Oscillation criteria for third order delay nonlinear differential equations

Directory of Open Access Journals (Sweden)

E. M. Elabbasy

2012-01-01

via comparison with some first differential equations whose oscillatory characters are known. Our results generalize and improve some known results for oscillation of third order nonlinear differential equations. Some examples are given to illustrate the main results.

Lag synchronization of chaotic systems with time-delayed linear

Indian Academy of Sciences (India)

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.

Maximum principle for a stochastic delayed system involving terminal state constraints.

Science.gov (United States)

Wen, Jiaqiang; Shi, Yufeng

2017-01-01

We investigate a stochastic optimal control problem where the controlled system is depicted as a stochastic differential delayed equation; however, at the terminal time, the state is constrained in a convex set. We firstly introduce an equivalent backward delayed system depicted as a time-delayed backward stochastic differential equation. Then a stochastic maximum principle is obtained by virtue of Ekeland's variational principle. Finally, applications to a state constrained stochastic delayed linear-quadratic control model and a production-consumption choice problem are studied to illustrate the main obtained result.

Nonlinear integrodifferential equations as discrete systems

Science.gov (United States)

Tamizhmani, K. M.; Satsuma, J.; Grammaticos, B.; Ramani, A.

1999-06-01

We analyse a class of integrodifferential equations of the `intermediate long wave' (ILW) type. We show that these equations can be formally interpreted as discrete, differential-difference systems. This allows us to link equations of this type with previous results of ours involving differential-delay equations and, on the basis of this, propose new integrable equations of ILW type. Finally, we extend this approach to pure difference equations and propose ILW forms for the discrete lattice KdV equation.

Modelling delays in pharmacokinetics

International Nuclear Information System (INIS)

Farooqi, Z.H.; Lambrecht, R.M.

1990-01-01

Linear system analysis has come to form the backbone of pharmacokinetics. Natural systems usually involve time delays, thus models incorporating them would be an order closer approximation to the real world compared to those that do not. Delays may be modelled in several ways. The approach considered in this study is to have a discrete-time delay dependent rate with the delay respresenting the duration between the entry of a drug into a compartment and its release in some form (may be as a metabolite) from the compartment. Such a delay may be because of one or more of several physiological reasons, like, formation of a reservoir, slow metabolism, or receptor binding. The mathematical structure this gives rise to is a system of delay-differential equations. Examples are given of simple one and two compartment systems with drugs like bumetanide, carbamazepine, and quinolone-caffeine interaction. In these examples generally a good fit is obtained and the suggested models form a good approximation. 21 refs., 6 figs

Constructing Hopf bifurcation lines for the stability of nonlinear systems with two time delays

Science.gov (United States)

Nguimdo, Romain Modeste

2018-03-01

Although the plethora real-life systems modeled by nonlinear systems with two independent time delays, the algebraic expressions for determining the stability of their fixed points remain the Achilles' heel. Typically, the approach for studying the stability of delay systems consists in finding the bifurcation lines separating the stable and unstable parameter regions. This work deals with the parametric construction of algebraic expressions and their use for the determination of the stability boundaries of fixed points in nonlinear systems with two independent time delays. In particular, we concentrate on the cases for which the stability of the fixed points can be ascertained from a characteristic equation corresponding to that of scalar two-delay differential equations, one-component dual-delay feedback, or nonscalar differential equations with two delays for which the characteristic equation for the stability analysis can be reduced to that of a scalar case. Then, we apply our obtained algebraic expressions to identify either the parameter regions of stable microwaves generated by dual-delay optoelectronic oscillators or the regions of amplitude death in identical coupled oscillators.

Numerical Bifurcation Analysis of Delayed Recycle Stream in a Continuously Stirred Tank Reactor

Science.gov (United States)

Gangadhar, Nalwala Rohitbabu; Balasubramanian, Periyasamy

2010-10-01

In this paper, we present the stability analysis of delay differential equations which arise as a result of transportation lag in the CSTR-mechanical separator recycle system. A first order irreversible elementary reaction is considered to model the system and is governed by the delay differential equations. The DDE-BIFTOOL software package is used to analyze the stability of the delay system. The present analysis reveals that the system exhibits delay independent stability for isothermal operation of the CSTR. In the absence of delay, the system is dynamically unstable for non-isothermal operation of the CSTR, and as a result of delay, the system exhibits delay dependent stability.

Mean, covariance, and effective dimension of stochastic distributed delay dynamics

Science.gov (United States)

René, Alexandre; Longtin, André

2017-11-01

Dynamical models are often required to incorporate both delays and noise. However, the inherently infinite-dimensional nature of delay equations makes formal solutions to stochastic delay differential equations (SDDEs) challenging. Here, we present an approach, similar in spirit to the analysis of functional differential equations, but based on finite-dimensional matrix operators. This results in a method for obtaining both transient and stationary solutions that is directly amenable to computation, and applicable to first order differential systems with either discrete or distributed delays. With fewer assumptions on the system's parameters than other current solution methods and no need to be near a bifurcation, we decompose the solution to a linear SDDE with arbitrary distributed delays into natural modes, in effect the eigenfunctions of the differential operator, and show that relatively few modes can suffice to approximate the probability density of solutions. Thus, we are led to conclude that noise makes these SDDEs effectively low dimensional, which opens the possibility of practical definitions of probability densities over their solution space.

Probabilistic delay differential equation modeling of event-related potentials.

Science.gov (United States)

Ostwald, Dirk; Starke, Ludger

2016-08-01

"Dynamic causal models" (DCMs) are a promising approach in the analysis of functional neuroimaging data due to their biophysical interpretability and their consolidation of functional-segregative and functional-integrative propositions. In this theoretical note we are concerned with the DCM framework for electroencephalographically recorded event-related potentials (ERP-DCM). Intuitively, ERP-DCM combines deterministic dynamical neural mass models with dipole-based EEG forward models to describe the event-related scalp potential time-series over the entire electrode space. Since its inception, ERP-DCM has been successfully employed to capture the neural underpinnings of a wide range of neurocognitive phenomena. However, in spite of its empirical popularity, the technical literature on ERP-DCM remains somewhat patchy. A number of previous communications have detailed certain aspects of the approach, but no unified and coherent documentation exists. With this technical note, we aim to close this gap and to increase the technical accessibility of ERP-DCM. Specifically, this note makes the following novel contributions: firstly, we provide a unified and coherent review of the mathematical machinery of the latent and forward models constituting ERP-DCM by formulating the approach as a probabilistic latent delay differential equation model. Secondly, we emphasize the probabilistic nature of the model and its variational Bayesian inversion scheme by explicitly deriving the variational free energy function in terms of both the likelihood expectation and variance parameters. Thirdly, we detail and validate the estimation of the model with a special focus on the explicit form of the variational free energy function and introduce a conventional nonlinear optimization scheme for its maximization. Finally, we identify and discuss a number of computational issues which may be addressed in the future development of the approach. Copyright © 2016 Elsevier Inc. All rights reserved.

Auto-Bäcklund transformations and special integrals for differential-delay Painlevé hierarchies

Science.gov (United States)

Fedorov, Yuri; Gordoa, Pilar R.; Pickering, Andrew

2014-10-01

The six Painlevé equations have attracted much interest over the last thirty years or so. More recently many authors have begun to explore properties of higher-order versions of both these equations and their discrete analogues. However, little attention has been paid to differential-delay Painlevé equations, i.e., analogues of the Painlevé equations involving both shifts in and derivatives with respect to the independent variable, and even less to higher-order analogues of these last. In the current paper we discuss the phenomenon whereby members of one differential-delay Painlevé hierarchy define solutions of higher-order members of a second differential-delay Painlevé hierarchy. We also give an auto-Bäcklund transformation for a differential-delay Painlevé hierarchy. The key to our approach is the underlying Hamiltonian structure of related completely integrable lattice hierarchies.

Stability and Hopf bifurcation in a delayed competitive web sites model

International Nuclear Information System (INIS)

Xiao Min; Cao Jinde

2006-01-01

The delayed differential equations modeling competitive web sites, based on the Lotka-Volterra competition equations, are considered. Firstly, the linear stability is investigated. It is found that there is a stability switch for time delay, and Hopf bifurcation occurs when time delay crosses through a critical value. Then the direction and stability of the bifurcated periodic solutions are determined, using the normal form theory and the center manifold reduction. Finally, some numerical simulations are carried out to illustrate the results found

Controllability of nonlinear delay oscillating systems

Directory of Open Access Journals (Sweden)

Chengbin Liang

2017-05-01

Full Text Available In this paper, we study the controllability of a system governed by second order delay differential equations. We introduce a delay Gramian matrix involving the delayed matrix sine, which is used to establish sufficient and necessary conditions of controllability for the linear problem. In addition, we also construct a specific control function for controllability. For the nonlinear problem, we construct a control function and transfer the controllability problem to a fixed point problem for a suitable operator. We give a sufficient condition to guarantee the nonlinear delay system is controllable. Two examples are given to illustrate our theoretical results by calculating a specific control function and inverse of a delay Gramian matrix.

Delay dynamical systems and applications to nonlinear machine-tool chatter

International Nuclear Information System (INIS)

Fofana, M.S.

2003-01-01

The stability behaviour of machine chatter that exhibits Hopf and degenerate bifurcations has been examined without the assumption of small delays between successive cuts. Delay dynamical system theory leading to the reduction of the infinite-dimensional character of the governing delay differential equations (DDEs) to a finite-dimensional set of ordinary differential equations have been employed. The essential mathematical arguments for these systems in the context of retarded DDEs are summarized. Then the application of these arguments in the stability study of machine-tool chatter with multiple time delays is presented. Explicit analytical expressions ensuring stable and unstable machining when perturbations are periodic, stochastic and nonlinear have been derived using the integral averaging method and Lyapunov exponents

Lyapunov functionals and stability of stochastic functional differential equations

CERN Document Server

Shaikhet, Leonid

2013-01-01

Stability conditions for functional differential equations can be obtained using Lyapunov functionals. Lyapunov Functionals and Stability of Stochastic Functional Differential Equations describes the general method of construction of Lyapunov functionals to investigate the stability of differential equations with delays. This work continues and complements the author’s previous book Lyapunov Functionals and Stability of Stochastic Difference Equations, where this method is described for discrete- and continuous-time difference equations. The text begins with a description of the peculiarities of deterministic and stochastic functional differential equations. There follow basic definitions for stability theory of stochastic hereditary systems, and a formal procedure of Lyapunov functionals construction is presented. Stability investigation is conducted for stochastic linear and nonlinear differential equations with constant and distributed delays. The proposed method is used for stability investigation of di...

Nonlinear Stability and Convergence of Two-Step Runge-Kutta Methods for Volterra Delay Integro-Differential Equations

Directory of Open Access Journals (Sweden)

Haiyan Yuan

2013-01-01

Full Text Available This paper introduces the stability and convergence of two-step Runge-Kutta methods with compound quadrature formula for solving nonlinear Volterra delay integro-differential equations. First, the definitions of (k,l-algebraically stable and asymptotically stable are introduced; then the asymptotical stability of a (k,l-algebraically stable two-step Runge-Kutta method with 0

Asymptotic Behaviour and Extinction of Delay Lotka-Volterra Model with Jump-Diffusion

OpenAIRE

Dan Li; Jing’an Cui; Guohua Song

2014-01-01

This paper studies the effect of jump-diffusion random environmental perturbations on the asymptotic behaviour and extinction of Lotka-Volterra population dynamics with delays. The contributions of this paper lie in the following: (a) to consider delay stochastic differential equation with jumps, we introduce a proper initial data space, in which the initial data may be discontinuous function with downward jumps; (b) we show that the delay stochastic differential equation with jumps associate...

Positioning in a flat two-dimensional space-time: The delay master equation

International Nuclear Information System (INIS)

Coll, Bartolome; Ferrando, Joan Josep; Morales-Lladosa, Juan Antonio

2010-01-01

The basic theory on relativistic positioning systems in a two-dimensional space-time has been presented in two previous papers [B. Coll, J. J. Ferrando, and J. A. Morales, Phys. Rev. D 73, 084017 (2006); ibid.74, 104003 (2006)], where the possibility of making relativistic gravimetry with these systems has been analyzed by considering specific examples. Here, generic relativistic positioning systems in the Minkowski plane are studied. The information that can be obtained from the data received by a user of the positioning system is analyzed in detail. In particular, it is shown that the accelerations of the emitters and of the user along their trajectories are determined by the sole knowledge of the emitter positioning data and of the acceleration of only one of the emitters. Moreover, as a consequence of the so-called master delay equation, the knowledge of this acceleration is only required during an echo interval, i.e., the interval between the emission time of a signal by an emitter and its reception time after being reflected by the other emitter. These results are illustrated with the obtention of the dynamics of the emitters and of the user from specific sets of data received by the user.

Neutron stochastic transport theory with delayed neutrons

International Nuclear Information System (INIS)

Munoz-Cobo, J.L.; Verdu, G.

1987-01-01

From the stochastic transport theory with delayed neutrons, the Boltzmann transport equation with delayed neutrons for the average flux emerges in a natural way without recourse to any approximation. From this theory a general expression is obtained for the Feynman Y-function when delayed neutrons are included. The single mode approximation for the particular case of a subcritical assembly is developed, and it is shown that Y-function reduces to the familiar expression quoted in many books, when delayed neutrons are not considered, and spatial and source effects are not included. (author)

Reconstruction of ensembles of coupled time-delay systems from time series.

Science.gov (United States)

Sysoev, I V; Prokhorov, M D; Ponomarenko, V I; Bezruchko, B P

2014-06-01

We propose a method to recover from time series the parameters of coupled time-delay systems and the architecture of couplings between them. The method is based on a reconstruction of model delay-differential equations and estimation of statistical significance of couplings. It can be applied to networks composed of nonidentical nodes with an arbitrary number of unidirectional and bidirectional couplings. We test our method on chaotic and periodic time series produced by model equations of ensembles of diffusively coupled time-delay systems in the presence of noise, and apply it to experimental time series obtained from electronic oscillators with delayed feedback coupled by resistors.

Delay-induced wave instabilities in single-species reaction-diffusion systems

Science.gov (United States)

Otto, Andereas; Wang, Jian; Radons, Günter

2017-11-01

The Turing (wave) instability is only possible in reaction-diffusion systems with more than one (two) components. Motivated by the fact that a time delay increases the dimension of a system, we investigate the presence of diffusion-driven instabilities in single-species reaction-diffusion systems with delay. The stability of arbitrary one-component systems with a single discrete delay, with distributed delay, or with a variable delay is systematically analyzed. We show that a wave instability can appear from an equilibrium of single-species reaction-diffusion systems with fluctuating or distributed delay, which is not possible in similar systems with constant discrete delay or without delay. More precisely, we show by basic analytic arguments and by numerical simulations that fast asymmetric delay fluctuations or asymmetrically distributed delays can lead to wave instabilities in these systems. Examples, for the resulting traveling waves are shown for a Fisher-KPP equation with distributed delay in the reaction term. In addition, we have studied diffusion-induced instabilities from homogeneous periodic orbits in the same systems with variable delay, where the homogeneous periodic orbits are attracting resonant periodic solutions of the system without diffusion, i.e., periodic orbits of the Hutchinson equation with time-varying delay. If diffusion is introduced, standing waves can emerge whose temporal period is equal to the period of the variable delay.

Synchronization in networks with heterogeneous coupling delays

Science.gov (United States)

Otto, Andreas; Radons, Günter; Bachrathy, Dániel; Orosz, Gábor

2018-01-01

Synchronization in networks of identical oscillators with heterogeneous coupling delays is studied. A decomposition of the network dynamics is obtained by block diagonalizing a newly introduced adjacency lag operator which contains the topology of the network as well as the corresponding coupling delays. This generalizes the master stability function approach, which was developed for homogenous delays. As a result the network dynamics can be analyzed by delay differential equations with distributed delay, where different delay distributions emerge for different network modes. Frequency domain methods are used for the stability analysis of synchronized equilibria and synchronized periodic orbits. As an example, the synchronization behavior in a system of delay-coupled Hodgkin-Huxley neurons is investigated. It is shown that the parameter regions where synchronized periodic spiking is unstable expand when increasing the delay heterogeneity.

Langevin approach to synchronization of hyperchaotic time-delay dynamics

Energy Technology Data Exchange (ETDEWEB)

Budini, Adrian A [Consejo Nacional de Investigaciones CientIficas y Tecnicas, Centro Atomico Bariloche, Av. E Bustillo Km 9.5, (8400) Bariloche (Argentina); Consortium of the Americas for Interdisciplinary Science and Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131 (United States)

2008-11-07

In this paper, we characterize the synchronization phenomenon of hyperchaotic scalar nonlinear delay dynamics in a fully-developed chaos regime. Our results rely on the observation that, in that regime, the stationary statistical properties of a class of hyperchaotic attractors can be reproduced with a linear Langevin equation, defined by replacing the nonlinear delay force by a delta-correlated noise. Therefore, the synchronization phenomenon can be analytically characterized by a set of coupled Langevin equations. We apply this formalism to study anticipated synchronization dynamics subject to external noise fluctuations as well as for characterizing the effects of parameter mismatch in a hyperchaotic communication scheme. The same procedure is applied to second-order differential delay equations associated with synchronization in electro-optical devices. In all cases, the departure with respect to perfect synchronization is measured through a similarity function. Numerical simulations in discrete maps associated with the hyperchaotic dynamics support the formalism.

The diffusive Lotka-Volterra predator-prey system with delay.

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Al Noufaey, K S; Marchant, T R; Edwards, M P

2015-12-01

Semi-analytical solutions for the diffusive Lotka-Volterra predator-prey system with delay are considered in one and two-dimensional domains. The Galerkin method is applied, which approximates the spatial structure of both the predator and prey populations. This approach is used to obtain a lower-order, ordinary differential delay equation model for the system of governing delay partial differential equations. Steady-state and transient solutions and the region of parameter space, in which Hopf bifurcations occur, are all found. In some cases simple linear expressions are found as approximations, to describe steady-state solutions and the Hopf parameter regions. An asymptotic analysis for the periodic solution near the Hopf bifurcation point is performed for the one-dimensional domain. An excellent agreement is shown in comparisons between semi-analytical and numerical solutions of the governing equations. Copyright © 2015 Elsevier Inc. All rights reserved.

Noise transmission and delay-induced stochasticoscillations in biochemical network motifs

Institute of Scientific and Technical Information of China (English)

Liu Sheng-Jun; Wang Qi; Liu Bo; Yan Shi-Wei; Fumihiko Sakata

2011-01-01

With the aid of stochastic delayed-feedback differential equations,we derive an analytic expression for the power spectra of reacting molecules included in a generic biological network motif that is incorporated with a feedback mechanism and time delays in gene regulation.We systematically analyse the effects of time delays,the feedback mechanism,and biological stochasticity on the power spectra.It has been clarified that the time delays together with the feedback mechanism can induce stochastic oscillations at the molecular level and invalidate the noise addition rule for a modular description of the noise propagator.Delay-induced stochastic resonance can be expected,which is related to the stability loss of the reaction systems and Hopf bifurcation occurring for solutions of the corresponding deterministic reaction equations.Through the analysis of the power spectrum,a new approach is proposed to estimate the oscillation period.

Stochastic two-delay differential model of delayed visual feedback effects on postural dynamics.

Science.gov (United States)

Boulet, Jason; Balasubramaniam, Ramesh; Daffertshofer, Andreas; Longtin, André

2010-01-28

We report on experiments and modelling involving the 'visuo-postural control loop' in the upright stance. We experimentally manipulated an artificial delay to the visual feedback during standing, presented at delays ranging from 0 to 1 s in increments of 250 ms. Using stochastic delay differential equations, we explicitly modelled the centre-of-pressure (COP) and centre-of-mass (COM) dynamics with two independent delay terms for vision and proprioception. A novel 'drifting fixed point' hypothesis was used to describe the fluctuations of the COM with the COP being modelled as a faster, corrective process of the COM. The model was in good agreement with the data in terms of probability density functions, power spectral densities, short- and long-term correlations (Hurst exponents) as well the critical time between the two ranges. This journal is © 2010 The Royal Society

Asymptotic analysis for functional stochastic differential equations

CERN Document Server

Bao, Jianhai; Yuan, Chenggui

2016-01-01

This brief treats dynamical systems that involve delays and random disturbances. The study is motivated by a wide variety of systems in real life in which random noise has to be taken into consideration and the effect of delays cannot be ignored. Concentrating on such systems that are described by functional stochastic differential equations, this work focuses on the study of large time behavior, in particular, ergodicity. This brief is written for probabilists, applied mathematicians, engineers, and scientists who need to use delay systems and functional stochastic differential equations in their work. Selected topics from the brief can also be used in a graduate level topics course in probability and stochastic processes.

Theoretical foundations for traditional and generalized sensitivity functions for nonlinear delay differential equations.

Science.gov (United States)

Banks, H Thomas; Robbins, Danielle; Sutton, Karyn L

2013-01-01

In this paper we present new results for differentiability of delay systems with respect to initial conditions and delays. After motivating our results with a wide range of delay examples arising in biology applications, we further note the need for sensitivity functions (both traditional and generalized sensitivity functions), especially in control and estimation problems. We summarize general existence and uniqueness results before turning to our main results on differentiation with respect to delays, etc. Finally we discuss use of our results in the context of estimation problems.

Are Gestational Age, Birth Weight, and Birth Length Indicators of Favorable Fetal Growth Conditions? A Structural Equation Analysis of Filipino Infants

OpenAIRE

Bollen, Kenneth A.; Noble, Mark D.; Adair, Linda S.

2013-01-01

The fetal origin hypothesis emphasizes the life-long health impacts of prenatal conditions. Birth weight, birth length, and gestational age are indicators of the fetal environment. However, these variables often have missing data and are subject to random and systematic errors caused by delays in measurement, differences in measurement instruments, and human error. With data from the Cebu (Philippines) Longitudinal Health and Nutrition Survey, we use structural equation models (SEMs), to expl...

Time-delayed feedback control of diffusion in random walkers

Science.gov (United States)

Ando, Hiroyasu; Takehara, Kohta; Kobayashi, Miki U.

2017-07-01

Time delay in general leads to instability in some systems, while specific feedback with delay can control fluctuated motion in nonlinear deterministic systems to a stable state. In this paper, we consider a stochastic process, i.e., a random walk, and observe its diffusion phenomenon with time-delayed feedback. As a result, the diffusion coefficient decreases with increasing delay time. We analytically illustrate this suppression of diffusion by using stochastic delay differential equations and justify the feasibility of this suppression by applying time-delayed feedback to a molecular dynamics model.

Convergence and stability of the exponential Euler method for semi-linear stochastic delay differential equations.

Science.gov (United States)

Zhang, Ling

2017-01-01

The main purpose of this paper is to investigate the strong convergence and exponential stability in mean square of the exponential Euler method to semi-linear stochastic delay differential equations (SLSDDEs). It is proved that the exponential Euler approximation solution converges to the analytic solution with the strong order [Formula: see text] to SLSDDEs. On the one hand, the classical stability theorem to SLSDDEs is given by the Lyapunov functions. However, in this paper we study the exponential stability in mean square of the exact solution to SLSDDEs by using the definition of logarithmic norm. On the other hand, the implicit Euler scheme to SLSDDEs is known to be exponentially stable in mean square for any step size. However, in this article we propose an explicit method to show that the exponential Euler method to SLSDDEs is proved to share the same stability for any step size by the property of logarithmic norm.

Convergence and stability of the exponential Euler method for semi-linear stochastic delay differential equations

Directory of Open Access Journals (Sweden)

Ling Zhang

2017-10-01

Full Text Available Abstract The main purpose of this paper is to investigate the strong convergence and exponential stability in mean square of the exponential Euler method to semi-linear stochastic delay differential equations (SLSDDEs. It is proved that the exponential Euler approximation solution converges to the analytic solution with the strong order 1 2 $\\frac{1}{2}$ to SLSDDEs. On the one hand, the classical stability theorem to SLSDDEs is given by the Lyapunov functions. However, in this paper we study the exponential stability in mean square of the exact solution to SLSDDEs by using the definition of logarithmic norm. On the other hand, the implicit Euler scheme to SLSDDEs is known to be exponentially stable in mean square for any step size. However, in this article we propose an explicit method to show that the exponential Euler method to SLSDDEs is proved to share the same stability for any step size by the property of logarithmic norm.

Bifurcation Analysis and Spatiotemporal Patterns in Unidirectionally Delay-Coupled Vibratory Gyroscopes

Science.gov (United States)

Li, Li; Xu, Jian

Time delay is inevitable in unidirectionally coupled drive-free vibratory gyroscope system. The effect of time delay on the gyroscope system is studied in this paper. To this end, amplitude death and Hopf bifurcation induced by small time delay are first investigated by analyzing the related characteristic equation. Then, the direction of Hopf bifurcations and stability of Hopf-bifurcating periodic oscillations are determined by calculating the normal form on the center manifold. Next, spatiotemporal patterns of these Hopf-bifurcating periodic oscillations are analyzed by using the symmetric bifurcation theory of delay differential equations. Finally, it is found that numerical simulations agree with the associated analytic results. These phenomena could be induced although time delay is very small. Therefore, it is shown that time delay is an important factor which influences the sensitivity and accuracy of the gyroscope system and cannot be neglected during the design and manufacture.

Parametric Sensitivity Analysis of Oscillatory Delay Systems with an Application to Gene Regulation.

Science.gov (United States)

Ingalls, Brian; Mincheva, Maya; Roussel, Marc R

2017-07-01

A parametric sensitivity analysis for periodic solutions of delay-differential equations is developed. Because phase shifts cause the sensitivity coefficients of a periodic orbit to diverge, we focus on sensitivities of the extrema, from which amplitude sensitivities are computed, and of the period. Delay-differential equations are often used to model gene expression networks. In these models, the parametric sensitivities of a particular genotype define the local geometry of the evolutionary landscape. Thus, sensitivities can be used to investigate directions of gradual evolutionary change. An oscillatory protein synthesis model whose properties are modulated by RNA interference is used as an example. This model consists of a set of coupled delay-differential equations involving three delays. Sensitivity analyses are carried out at several operating points. Comments on the evolutionary implications of the results are offered.

A new car-following model with two delays

International Nuclear Information System (INIS)

Yu, Lei; Shi, Zhong-ke; Li, Tong

2014-01-01

A new car-following model is proposed by taking into account two different time delays in sensing headway and velocity. The effect of time delays on the stability analysis is studied. The theoretical and numerical results show that traffic jams are suppressed efficiently when the difference between two time delays decreases and those can be described by the solution of the modified Korteweg–de Vries (mKdV) equation. Traffic flow is more stable with two delays in headway and velocity than in the case with only one delay in headway. The impact of local small disturbance to the system is also studied.

A Fast Time-Delay Calculation Method in Through-Wall-Radar Detection Scenario

Directory of Open Access Journals (Sweden)

Zhang Qi

2016-01-01

Full Text Available In TWR (Through Wall Radar signal processing procedure, time delay estimation is one of the key steps in target localization and high resolution imaging. In time domain imaging procedure such as back projection imaging algorithm, round trip propagation time delay at the path of “transmitter-target-receiver” needs to be calculated for each pixel in imaging region. In typical TWR scenario, transmitter and receiver are at one side and targets at the other side of a wall. Based on two-dimensional searching algorithm or solving two variables equation of four times, traditional time delay calculation algorithms are complex and time consuming, and cannot be used to real-time imaging procedure. In this paper, a new fast time-delay (FTD algorithm is presented. Because of that incident angle at one side equals to refracting angle at the other side, an equation of lateral distance through the wall can be established. By solving this equation, the lateral distance can be obtained and total propagation time delay can be calculated subsequently. Through processing simulation data, the result shows that new algorithm can be applied effectively to real-time time-delay calculation in TWR signal processing.

A Stochastic Delay Model For Pricing Debt And Loan Guarantees: Theoretical Results

OpenAIRE

Kemajou, Elisabeth; Mohammed, Salah-Eldin; Tambue, Antoine

2012-01-01

We consider that the price of a firm follows a non linear stochastic delay differential equation. We also assume that any claim value whose value depends on firm value and time follows a non linear stochastic delay differential equation. Using self-financed strategy and replication we are able to derive a Random Partial Differential Equation (RPDE) satisfied by any corporate claim whose value is a function of firm value and time. Under specific final and boundary conditions, we solve the RPDE...

A differential transformation approach for solving functional differential equations with multiple delays

Science.gov (United States)

Rebenda, Josef; Šmarda, Zdeněk

2017-07-01

In the paper an efficient semi-analytical approach based on the method of steps and the differential transformation is proposed for numerical approximation of solutions of functional differential models of delayed and neutral type on a finite interval of arbitrary length, including models with several constant delays. Algorithms for both commensurate and non-commensurate delays are described, applications are shown in examples. Validity and efficiency of the presented algorithms is compared with the variational iteration method, the Adomian decomposition method and the polynomial least squares method numerically. Matlab package DDE23 is used to produce reference numerical values.

Asymptotics of a Steady-State Condition of Finite-Difference Approximation of a Logistic Equation with Delay and Small Diffusion

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S. A. Kaschenko

2014-01-01

Full Text Available We study the dynamics of finite-difference approximation on spatial variables of a logistic equation with delay and diffusion. It is assumed that the diffusion coefficient is small and the Malthusian coefficient is large. The question of the existence and asymptotic behavior of attractors was studied with special asymptotic methods. It is shown that there is a rich array of different types of attractors in the phase space: leading centers, spiral waves, etc. The main asymptotic characteristics of all solutions from the corresponding attractors are adduced in this work. Typical graphics of wave fronts motion of different structures are represented in the article.

Dynamics of Nonlinear Time-Delay Systems

CERN Document Server

Lakshmanan, Muthusamy

2010-01-01

Synchronization of chaotic systems, a patently nonlinear phenomenon, has emerged as a highly active interdisciplinary research topic at the interface of physics, biology, applied mathematics and engineering sciences. In this connection, time-delay systems described by delay differential equations have developed as particularly suitable tools for modeling specific dynamical systems. Indeed, time-delay is ubiquitous in many physical systems, for example due to finite switching speeds of amplifiers in electronic circuits, finite lengths of vehicles in traffic flows, finite signal propagation times in biological networks and circuits, and quite generally whenever memory effects are relevant. This monograph presents the basics of chaotic time-delay systems and their synchronization with an emphasis on the effects of time-delay feedback which give rise to new collective dynamics. Special attention is devoted to scalar chaotic/hyperchaotic time-delay systems, and some higher order models, occurring in different bran...

Global attractivity of positive periodic solution to periodic Lotka-Volterra competition systems with pure delay

Science.gov (United States)

Tang, Xianhua; Cao, Daomin; Zou, Xingfu

We consider a periodic Lotka-Volterra competition system without instantaneous negative feedbacks (i.e., pure-delay systems) x(t)=x(t)[r(t)-∑j=1na(t)x(t-τ(t))], i=1,2,…,n. We establish some 3/2-type criteria for global attractivity of a positive periodic solution of the system, which generalize the well-known Wright's 3/2 criteria for the autonomous delay logistic equation, and thereby, address the open problem proposed by both Kuang [Y. Kuang, Global stability in delayed nonautonomous Lotka-Volterra type systems without saturated equilibria, Differential Integral Equations 9 (1996) 557-567] and Teng [Z. Teng, Nonautonomous Lotka-Volterra systems with delays, J. Differential Equations 179 (2002) 538-561].

A delay differential equation model for dengue transmission with regular visits to a mosquito breeding site

Science.gov (United States)

Yaacob, Y.; Yeak, S. H.; Lim, R. S.; Soewono, E.

2015-03-01

Dengue disease has been known as one of widely transmitted vector-borne diseases which potentially affects millions of people throughout the world especially in tropical and sub-tropical countries. One of the main factors contributing in the complication of the transmission process is the mobility of people in which people may get infection in the places far from their home. Here we construct a delay differential equation model for dengue transmission in a closed population where regular visits of people to a mosquito breeding site out of their residency such as traditional market take place daily. Basic reproductive ratio of the system is obtained and depends on the ratio between the outgoing rates of susceptible human and infective human. It is shown that the increase of mobility with different variation of mobility rates may contribute to different level of basic reproductive ratio as well as different level of outbreaks.

Multi-response model for rheumatoid arthritis based on delay differential equations in collagen-induced arthritic mice treated with an anti-GM-CSF antibody.

Science.gov (United States)

Koch, Gilbert; Wagner, Thomas; Plater-Zyberk, Christine; Lahu, Gezim; Schropp, Johannes

2012-02-01

Collagen-induced arthritis (CIA) in mice is an experimental model for rheumatoid arthritis, a human chronic inflammatory destructive disease. The therapeutic effect of neutralizing the cytokine granulocyte-macrophage colony-stimulating factor (GM-CSF) by an antibody was examined in the mouse disease in a view of deriving a pharmacokinetic/pharmacodynamic (PKPD) model. In CIA mice the development of disease is measured by a total arthritic score (TAS) and an ankylosis score (AKS). We present a multi-response PKPD model which describes the time course of the unperturbed and perturbed TAS and AKS. The antibody acts directly on GM-CSF by binding to it. Therefore, a compartment for the cytokine GM-CSF is an essential component of the mathematical model. This compartment drives the disease development in the PKPD model. Different known properties of arthritis development in the CIA model are included in the PKPD model. Firstly, the inflammation, driven by GM-CSF, dominates at the beginning of the disease and decreases after some time. Secondly, a destructive (ankylosis) part evolves in the TAS that is delayed in time. In order to model these two properties a delay differential equation was used. The PKPD model was applied to different experiments with doses ranging from 0.1 to 100 mg/kg. The influence of the drug was modeled by a non-linear approach. The final mathematical model consists of three differential equations representing the compartments for GM-CSF, inflammation and destruction. Our mathematical model described well all available dosing schedules by a simultaneous fit. We also present an equivalent and easy reformulation as ordinary differential equation which grants the use of standard PKPD software.

A continuous time Cournot duopoly with delays

International Nuclear Information System (INIS)

Gori, Luca; Guerrini, Luca; Sodini, Mauro

2015-01-01

This paper extends the classical repeated duopoly model with quantity-setting firms of Bischi et al. (1998) by assuming that production of goods is subject to some gestation lags but exchanges take place continuously in the market. The model is expressed in the form of differential equations with discrete delays. By using some recent mathematical techniques and numerical experiments, results show some dynamic phenomena that cannot be observed when delays are absent. In addition, depending on the extent of time delays and inertia, synchronisation failure can arise even in the event of homogeneous firms.

Noise transmission and delay-induced stochastic oscillations in biochemical network motifs

International Nuclear Information System (INIS)

Liu Sheng-Jun; Wang Qi; Liu Bo; Yan Shi-Wei; Sakata Fumihiko

2011-01-01

With the aid of stochastic delayed-feedback differential equations, we derive an analytic expression for the power spectra of reacting molecules included in a generic biological network motif that is incorporated with a feedback mechanism and time delays in gene regulation. We systematically analyse the effects of time delays, the feedback mechanism, and biological stochasticity on the power spectra. It has been clarified that the time delays together with the feedback mechanism can induce stochastic oscillations at the molecular level and invalidate the noise addition rule for a modular description of the noise propagator. Delay-induced stochastic resonance can be expected, which is related to the stability loss of the reaction systems and Hopf bifurcation occurring for solutions of the corresponding deterministic reaction equations. Through the analysis of the power spectrum, a new approach is proposed to estimate the oscillation period. (interdisciplinary physics and related areas of science and technology)

Deriving the bond pricing equation

Directory of Open Access Journals (Sweden)

Kožul Nataša

2014-01-01

Full Text Available Given the recent focus on Eurozone debt crisis and the credit rating downgrade not only of US debt, but that of other countries and many UK major banking institutions, this paper aims to explain the concept of bond yield, its different measures and bond pricing equation. Yields on capital market instruments are rarely quoted on the same basis, which makes direct comparison between different as investment choices impossible. Some debt instruments are quoted on discount basis, whilst coupon-bearing ones accrue interest differently, offer different compounding opportunities, have different coupon payment frequencies, and manage non-business day maturity dates differently. Moreover, rules governing debt vary across countries, markets and currencies, making yield calculation and comparison a rather complex issue. Thus, some fundamental concepts applicable to debt instrument yield measurement, with focus on bond equation, are presented here. In addition, bond equation expressed in annuity form and used to apply Newton-Raphson algorithm to derive true bond yield is also shown.

Error characterization for asynchronous computations: Proxy equation approach

Science.gov (United States)

Sallai, Gabriella; Mittal, Ankita; Girimaji, Sharath

2017-11-01

Numerical techniques for asynchronous fluid flow simulations are currently under development to enable efficient utilization of massively parallel computers. These numerical approaches attempt to accurately solve time evolution of transport equations using spatial information at different time levels. The truncation error of asynchronous methods can be divided into two parts: delay dependent (EA) or asynchronous error and delay independent (ES) or synchronous error. The focus of this study is a specific asynchronous error mitigation technique called proxy-equation approach. The aim of this study is to examine these errors as a function of the characteristic wavelength of the solution. Mitigation of asynchronous effects requires that the asynchronous error be smaller than synchronous truncation error. For a simple convection-diffusion equation, proxy-equation error analysis identifies critical initial wave-number, λc. At smaller wave numbers, synchronous error are larger than asynchronous errors. We examine various approaches to increase the value of λc in order to improve the range of applicability of proxy-equation approach.

Finger tapping movements of Parkinson's disease patients automatically rated using nonlinear delay differential equations.

Science.gov (United States)

Lainscsek, C; Rowat, P; Schettino, L; Lee, D; Song, D; Letellier, C; Poizner, H

2012-03-01

Parkinson's disease is a degenerative condition whose severity is assessed by clinical observations of motor behaviors. These are performed by a neurological specialist through subjective ratings of a variety of movements including 10-s bouts of repetitive finger-tapping movements. We present here an algorithmic rating of these movements which may be beneficial for uniformly assessing the progression of the disease. Finger-tapping movements were digitally recorded from Parkinson's patients and controls, obtaining one time series for every 10 s bout. A nonlinear delay differential equation, whose structure was selected using a genetic algorithm, was fitted to each time series and its coefficients were used as a six-dimensional numerical descriptor. The algorithm was applied to time-series from two different groups of Parkinson's patients and controls. The algorithmic scores compared favorably with the unified Parkinson's disease rating scale scores, at least when the latter adequately matched with ratings from the Hoehn and Yahr scale. Moreover, when the two sets of mean scores for all patients are compared, there is a strong (r = 0.785) and significant (p<0.0015) correlation between them.

Exponential mean-square stability of two classes of theta Milstein methods for stochastic delay differential equations

Science.gov (United States)

Rouz, Omid Farkhondeh; Ahmadian, Davood; Milev, Mariyan

2017-12-01

This paper establishes exponential mean square stability of two classes of theta Milstein methods, namely split-step theta Milstein (SSTM) method and stochastic theta Milstein (STM) method, for stochastic differential delay equations (SDDEs). We consider the SDDEs problem under a coupled monotone condition on drift and diffusion coefficients, as well as a necessary linear growth condition on the last term of theta Milstein method. It is proved that the SSTM method with θ ∈ [0, ½] can recover the exponential mean square stability of the exact solution with some restrictive conditions on stepsize, but for θ ∈ (½, 1], we proved that the stability results hold for any stepsize. Then, based on the stability results of SSTM method, we examine the exponential mean square stability of the STM method and obtain the similar stability results to that of the SSTM method. In the numerical section the figures show thevalidity of our claims.

Dynamics in a delayed-neural network

International Nuclear Information System (INIS)

Yuan Yuan

2007-01-01

In this paper, we consider a neural network of four identical neurons with time-delayed connections. Some parameter regions are given for global, local stability and synchronization using the theory of functional differential equations. The root distributions in the corresponding characteristic transcendental equation are analyzed, Pitchfork bifurcation, Hopf and equivariant Hopf bifurcations are investigated by revealing the center manifolds and normal forms. Numerical simulations are shown the agreements with the theoretical results

Instability in time-delayed switched systems induced by fast and random switching

Science.gov (United States)

Guo, Yao; Lin, Wei; Chen, Yuming; Wu, Jianhong

2017-07-01

In this paper, we consider a switched system comprising finitely or infinitely many subsystems described by linear time-delayed differential equations and a rule that orchestrates the system switching randomly among these subsystems, where the switching times are also randomly chosen. We first construct a counterintuitive example where even though all the time-delayed subsystems are exponentially stable, the behaviors of the randomly switched system change from stable dynamics to unstable dynamics with a decrease of the dwell time. Then by using the theories of stochastic processes and delay differential equations, we present a general result on when this fast and random switching induced instability should occur and we extend this to the case of nonlinear time-delayed switched systems as well.

Distributed Optimal Consensus Control for Multiagent Systems With Input Delay.

Science.gov (United States)

Zhang, Huaipin; Yue, Dong; Zhao, Wei; Hu, Songlin; Dou, Chunxia; Huaipin Zhang; Dong Yue; Wei Zhao; Songlin Hu; Chunxia Dou; Hu, Songlin; Zhang, Huaipin; Dou, Chunxia; Yue, Dong; Zhao, Wei

2018-06-01

This paper addresses the problem of distributed optimal consensus control for a continuous-time heterogeneous linear multiagent system subject to time varying input delays. First, by discretization and model transformation, the continuous-time input-delayed system is converted into a discrete-time delay-free system. Two delicate performance index functions are defined for these two systems. It is shown that the performance index functions are equivalent and the optimal consensus control problem of the input-delayed system can be cast into that of the delay-free system. Second, by virtue of the Hamilton-Jacobi-Bellman (HJB) equations, an optimal control policy for each agent is designed based on the delay-free system and a novel value iteration algorithm is proposed to learn the solutions to the HJB equations online. The proposed adaptive dynamic programming algorithm is implemented on the basis of a critic-action neural network (NN) structure. Third, it is proved that local consensus errors of the two systems and weight estimation errors of the critic-action NNs are uniformly ultimately bounded while the approximated control policies converge to their target values. Finally, two simulation examples are presented to illustrate the effectiveness of the developed method.

Delay-independent stability of genetic regulatory networks.

Science.gov (United States)

Wu, Fang-Xiang

2011-11-01

Genetic regulatory networks can be described by nonlinear differential equations with time delays. In this paper, we study both locally and globally delay-independent stability of genetic regulatory networks, taking messenger ribonucleic acid alternative splicing into consideration. Based on nonnegative matrix theory, we first develop necessary and sufficient conditions for locally delay-independent stability of genetic regulatory networks with multiple time delays. Compared to the previous results, these conditions are easy to verify. Then we develop sufficient conditions for global delay-independent stability for genetic regulatory networks. Compared to the previous results, this sufficient condition is less conservative. To illustrate theorems developed in this paper, we analyze delay-independent stability of two genetic regulatory networks: a real-life repressilatory network with three genes and three proteins, and a synthetic gene regulatory network with five genes and seven proteins. The simulation results show that the theorems developed in this paper can effectively determine the delay-independent stability of genetic regulatory networks.

Statistics and dimension of chaos in differential delay systems

Energy Technology Data Exchange (ETDEWEB)

Dorizzi, B.; Grammaticos, B.; Le Berre, M.; Pomeau, Y.; Ressayre, E.; Tallet, A.

1987-01-01

The chaotic solution of dissipative scalar-delay-differential equations with a nonlinear feedback periodic with respect to its argument is shown to behave as a Gaussian-Markovian process in a large time scale. The short time scale is shown to be defined by the correlation time of the delayed feedback. The dimension of the chaotic attractor is shown to be approximately equal to the number of short times that are contained inside the delay.

Statistics and dimension of chaos in differential delay systems

International Nuclear Information System (INIS)

Dorizzi, B.; Grammaticos, B.; Le Berre, M.; Pomeau, Y.; Ressayre, E.; Tallet, A.

1987-01-01

The chaotic solution of dissipative scalar-delay-differential equations with a nonlinear feedback periodic with respect to its argument is shown to behave as a Gaussian-Markovian process in a large time scale. The short time scale is shown to be defined by the correlation time of the delayed feedback. The dimension of the chaotic attractor is shown to be approximately equal to the number of short times that are contained inside the delay

Mean-Square Convergence of Drift-Implicit One-Step Methods for Neutral Stochastic Delay Differential Equations with Jump Diffusion

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Lin Hu

2011-01-01

Full Text Available A class of drift-implicit one-step schemes are proposed for the neutral stochastic delay differential equations (NSDDEs driven by Poisson processes. A general framework for mean-square convergence of the methods is provided. It is shown that under certain conditions global error estimates for a method can be inferred from estimates on its local error. The applicability of the mean-square convergence theory is illustrated by the stochastic θ-methods and the balanced implicit methods. It is derived from Theorem 3.1 that the order of the mean-square convergence of both of them for NSDDEs with jumps is 1/2. Numerical experiments illustrate the theoretical results. It is worth noting that the results of mean-square convergence of the stochastic θ-methods and the balanced implicit methods are also new.

Reduction theories elucidate the origins of complex biological rhythms generated by interacting delay-induced oscillations.

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Ikuhiro Yamaguchi

Full Text Available Time delay is known to induce sustained oscillations in many biological systems such as electroencephalogram (EEG activities and gene regulations. Furthermore, interactions among delay-induced oscillations can generate complex collective rhythms, which play important functional roles. However, due to their intrinsic infinite dimensionality, theoretical analysis of interacting delay-induced oscillations has been limited. Here, we show that the two primary methods for finite-dimensional limit cycles, namely, the center manifold reduction in the vicinity of the Hopf bifurcation and the phase reduction for weak interactions, can successfully be applied to interacting infinite-dimensional delay-induced oscillations. We systematically derive the complex Ginzburg-Landau equation and the phase equation without delay for general interaction networks. Based on the reduced low-dimensional equations, we demonstrate that diffusive (linearly attractive coupling between a pair of delay-induced oscillations can exhibit nontrivial amplitude death and multimodal phase locking. Our analysis provides unique insights into experimentally observed EEG activities such as sudden transitions among different phase-locked states and occurrence of epileptic seizures.

Stability Analysis for Fractional-Order Linear Singular Delay Differential Systems

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Hai Zhang

2014-01-01

Full Text Available We investigate the delay-independently asymptotic stability of fractional-order linear singular delay differential systems. Based on the algebraic approach, the sufficient conditions are presented to ensure the asymptotic stability for any delay parameter. By applying the stability criteria, one can avoid solving the roots of transcendental equations. An example is also provided to illustrate the effectiveness and applicability of the theoretical results.

Parameter estimation of a delay dynamical system using synchronization in presence of noise

International Nuclear Information System (INIS)

Rakshit, Biswambhar; Chowdhury, A. Roy; Saha, Papri

2007-01-01

A method of parameter estimation of a time delay chaotic system through synchronization is discussed. It is assumed that the observed data can always be effected with some white Gaussian noise. A least square approach is used to derive a system of differential equations which governs the temporal evolution of the parameters. These system of equations together with the coupled delay dynamical systems, when integrated, leads to asymptotic convergence to the value of the parameter along with synchronization of the two system variables. This method is quite effective for estimating the delay time which is an important characteristic feature of a delay dynamical system. The procedure is quite robust in the presence of noise

Positivity of Fundamental Matrix and Exponential Stability of Delay Differential System

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Alexander Domoshnitsky

2014-01-01

Full Text Available The classical Wazewski theorem established that nonpositivity of all nondiagonal elements pij  (i≠j,  i,j=1,…,n is necessary and sufficient for nonnegativity of the fundamental (Cauchy matrix and consequently for applicability of the Chaplygin approach of approximate integration for system of linear ordinary differential equations xi′t+∑j=1n‍pijtxjt=fit,   i=1,…,n. Results on nonnegativity of the Cauchy matrix for system of delay differential equations xi′t+∑j=1n‍pijtxjhijt=fit,   i=1,…,n, which were based on nonpositivity of all diagonal elements, were presented in the previous works. Then examples, which demonstrated that nonpositivity of nondiagonal coefficients pij is not necessary for systems of delay equations, were found. In this paper first sufficient results about nonnegativity of the Cauchy matrix of the delay system without this assumption are proven. A necessary condition of nonnegativity of the Cauchy matrix is proposed. On the basis of these results on nonnegativity of the Cauchy matrix, necessary and sufficient conditions of the exponential stability of the delay system are obtained.

Noise-sustained fluctuations in stochastic dynamics with a delay.

Science.gov (United States)

D'Odorico, Paolo; Laio, Francesco; Ridolfi, Luca

2012-04-01

Delayed responses to external drivers are ubiquitous in environmental, social, and biological processes. Delays may induce oscillations, Hopf bifurcations, and instabilities in deterministic systems even in the absence of nonlinearities. Despite recent advances in the study of delayed stochastic differential equations, the interaction of random drivers with delays remains poorly understood. In particular, it is unclear whether noise-induced behaviors may emerge from these interactions. Here we show that noise may enhance and sustain transient periodic oscillations inherent to deterministic delayed systems. We investigate the conditions conducive to the emergence and disappearance of these dynamics in a linear system in the presence of both additive and multiplicative noise.

Predictor feedback for delay systems implementations and approximations

CERN Document Server

Karafyllis, Iasson

2017-01-01

This monograph bridges the gap between the nonlinear predictor as a concept and as a practical tool, presenting a complete theory of the application of predictor feedback to time-invariant, uncertain systems with constant input delays and/or measurement delays. It supplies several methods for generating the necessary real-time solutions to the systems’ nonlinear differential equations, which the authors refer to as approximate predictors. Predictor feedback for linear time-invariant (LTI) systems is presented in Part I to provide a solid foundation on the necessary concepts, as LTI systems pose fewer technical difficulties than nonlinear systems. Part II extends all of the concepts to nonlinear time-invariant systems. Finally, Part III explores extensions of predictor feedback to systems described by integral delay equations and to discrete-time systems. The book’s core is the design of control and observer algorithms with which global stabilization, guaranteed in the previous literature with idealized (b...

Sharp bounds for periodic solutions of Lipschitzian differential equations

International Nuclear Information System (INIS)

Zevin, A A

2009-01-01

A general system of Lipschitzian differential equations, containing simultaneously terms without delay and with arbitrary constant and time-varying delays, is considered. For the autonomous case, a lower bound for the period of nonconstant periodic solutions, expressed in the respective supremum Lipschitz constants, is found. For nonautonomous periodic equations, explicit upper bounds for the amplitudes and maximum derivatives of periodic solutions are obtained. For all n ≥ 2, the bounds do not depend on n and, in general, are different from that for n = 1. All the bounds are sharp; they are attained in linear differential equations with piece-wise constant deviating arguments. A relation between the obtained bounds and the sharp bounds in other metrics is established

Dynamics of microbubble oscillators with delay coupling

Science.gov (United States)

Heckman, C. R.; Sah, S. M.; Rand, R. H.

2010-10-01

We investigate the stability of the in-phase mode in a system of two delay-coupled bubble oscillators. The bubble oscillator model is based on a 1956 paper by Keller and Kolodner. Delay coupling is due to the time it takes for a signal to travel from one bubble to another through the liquid medium that surrounds them. Using techniques from the theory of differential-delay equations as well as perturbation theory, we show that the equilibrium of the in-phase mode can be made unstable if the delay is long enough and if the coupling strength is large enough, resulting in a Hopf bifurcation. We then employ Lindstedt's method to compute the amplitude of the limit cycle as a function of the time delay. This work is motivated by medical applications involving noninvasive localized drug delivery via microbubbles.

Extinction and permanence in delayed stage-structure predator-prey system with impulsive effects

International Nuclear Information System (INIS)

Pang Guoping; Wang Fengyan; Chen Lansun

2009-01-01

Based on the classical stage-structured model and Lotka-Volterra predator-prey model, an impulsive delayed differential equation to model the process of periodically releasing natural enemies at fixed times for pest control is proposed and investigated. We show that the conditions for global attractivity of the 'pest-extinction' ('prey-eradication') periodic solution and permanence of the population of the model depend on time delay. We also show that constant maturation time delay and impulsive releasing for the predator can bring great effects on the dynamics of system by numerical analysis. As a result, the pest maturation time delay is considered to establish a procedure to maintain the pests at an acceptably low level in the long term. In this paper, the main feature is that we introduce time delay and pulse into the predator-prey (natural enemy-pest) model with age structure, exhibit a new modelling method which is applied to investigate impulsive delay differential equations, and give some reasonable suggestions for pest management.

Adaptive control of chaotic continuous-time systems with delay

Science.gov (United States)

Tian, Yu-Chu; Gao, Furong

1998-06-01

A simple delay system governed by a first-order differential-delay equation may behave chaotically, but the conditions for the system to have such behaviors have not been well recognized. In this paper, a set of rules is postulated first for the conditions for the delay system to display chaos. A model-reference adaptive control scheme is then proposed to control the chaotic system state to converge to an arbitrarily given reference trajectory with certain and uncertain system parameters. Numerical examples are given to analyze the chaotic behaviors of the delay system and to demonstrate the effectiveness of the proposed adaptive control scheme.

Relaxation Cycles in a Generalized Neuron Model with Two Delays

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S. D. Glyzin

2013-01-01

Full Text Available A method of modeling the phenomenon of bursting behavior in neural systems based on delay equations is proposed. A singularly perturbed scalar nonlinear differentialdifference equation of Volterra type is a mathematical model of a neuron and a separate pulse containing one function without delay and two functions with different lags. It is established that this equation, for a suitable choice of parameters, has a stable periodic motion with any preassigned number of bursts in the time interval of the period length. To prove this assertion we first go to a relay-type equation and then determine the asymptotic solutions of a singularly perturbed equation. On the basis of this asymptotics the Poincare operator is constructed. The resulting operator carries a closed bounded convex set of initial conditions into itself, which suggests that it has at least one fixed point. The Frechet derivative evaluation of the succession operator, made in the paper, allows us to prove the uniqueness and stability of the resulting relax of the periodic solution.

Multiple-parameter bifurcation analysis in a Kuramoto model with time delay and distributed shear

Science.gov (United States)

Niu, Ben; Zhang, Jiaming; Wei, Junjie

2018-05-01

In this paper, time delay effect and distributed shear are considered in the Kuramoto model. On the Ott-Antonsen's manifold, through analyzing the associated characteristic equation of the reduced functional differential equation, the stability boundary of the incoherent state is derived in multiple-parameter space. Moreover, very rich dynamical behavior such as stability switches inducing synchronization switches can occur in this equation. With the loss of stability, Hopf bifurcating coherent states arise, and the criticality of Hopf bifurcations is determined by applying the normal form theory and the center manifold theorem. On one hand, theoretical analysis indicates that the width of shear distribution and time delay can both eliminate the synchronization then lead the Kuramoto model to incoherence. On the other, time delay can induce several coexisting coherent states. Finally, some numerical simulations are given to support the obtained results where several bifurcation diagrams are drawn, and the effect of time delay and shear is discussed.

Noise-and delay-induced phase transitions of the dimer–monomer surface reaction model

International Nuclear Information System (INIS)

Zeng Chunhua; Wang Hua

2012-01-01

Highlights: ► We study the dimer–monomer surface reaction model. ► We show that noise induces first-order irreversible phase transition (IPT). ► Combination of noise and time-delayed feedback induce first- and second-order IPT. ► First- and second-order IPT is viewed as noise-and delay-induced phase transitions. - Abstract: The effects of noise and time-delayed feedback in the dimer–monomer (DM) surface reaction model are investigated. Applying small delay approximation, we construct a stochastic delayed differential equation and its Fokker–Planck equation to describe the state evolution of the DM reaction model. We show that the noise can only induce first-order irreversible phase transition (IPT) characteristic of the DM model, however the combination of the noise and time-delayed feedback can simultaneously induce first- and second-order IPT characteristics of the DM model. Therefore, it is shown that the well-known first- and second-order IPT characteristics of the DM model may be viewed as noise-and delay-induced phase transitions.

Heterogeneity of time delays determines synchronization of coupled oscillators.

Science.gov (United States)

Petkoski, Spase; Spiegler, Andreas; Proix, Timothée; Aram, Parham; Temprado, Jean-Jacques; Jirsa, Viktor K

2016-07-01

Network couplings of oscillatory large-scale systems, such as the brain, have a space-time structure composed of connection strengths and signal transmission delays. We provide a theoretical framework, which allows treating the spatial distribution of time delays with regard to synchronization, by decomposing it into patterns and therefore reducing the stability analysis into the tractable problem of a finite set of delay-coupled differential equations. We analyze delay-structured networks of phase oscillators and we find that, depending on the heterogeneity of the delays, the oscillators group in phase-shifted, anti-phase, steady, and non-stationary clusters, and analytically compute their stability boundaries. These results find direct application in the study of brain oscillations.

From dynamical systems with time-varying delay to circle maps and Koopman operators

Science.gov (United States)

Müller, David; Otto, Andreas; Radons, Günter

2017-06-01

In this paper, we investigate the influence of the retarded access by a time-varying delay on the dynamics of delay systems. We show that there are two universality classes of delays, which lead to fundamental differences in dynamical quantities such as the Lyapunov spectrum. Therefore, we introduce an operator theoretic framework, where the solution operator of the delay system is decomposed into the Koopman operator describing the delay access and an operator similar to the solution operator known from systems with constant delay. The Koopman operator corresponds to an iterated map, called access map, which is defined by the iteration of the delayed argument of the delay equation. The dynamics of this one-dimensional iterated map determines the universality classes of the infinite-dimensional state dynamics governed by the delay differential equation. In this way, we connect the theory of time-delay systems with the theory of circle maps and the framework of the Koopman operator. In this paper, we extend our previous work [A. Otto, D. Müller, and G. Radons, Phys. Rev. Lett. 118, 044104 (2017), 10.1103/PhysRevLett.118.044104] by elaborating the mathematical details and presenting further results also on the Lyapunov vectors.

Delayed feedback control of fractional-order chaotic systems

International Nuclear Information System (INIS)

Gjurchinovski, A; Urumov, V; Sandev, T

2010-01-01

We study the possibility to stabilize unstable steady states and unstable periodic orbits in chaotic fractional-order dynamical systems by the time-delayed feedback method. By performing a linear stability analysis, we establish the parameter ranges for successful stabilization of unstable equilibria in the plane parameterized by the feedback gain and the time delay. An insight into the control mechanism is gained by analyzing the characteristic equation of the controlled system, showing that the control scheme fails to control unstable equilibria having an odd number of positive real eigenvalues. We demonstrate that the method can also stabilize unstable periodic orbits for a suitable choice of the feedback gain, providing that the time delay is chosen to coincide with the period of the target orbit. In addition, it is shown numerically that delayed feedback control with a sinusoidally modulated time delay significantly enlarges the stability region of steady states in comparison to the classical time-delayed feedback scheme with a constant delay.

A multi scale approximation solution for the time dependent Boltzmann-transport equation

International Nuclear Information System (INIS)

Merk, B.

2004-03-01

The basis of all transient simulations for nuclear reactor cores is the reliable calculation of the power production. The local power distribution is generally calculated by solving the space, time, energy and angle dependent neutron transport equation known as Boltzmann equation. The computation of exact solutions of the Boltzmann equation is very time consuming. For practical numerical simulations approximated solutions are usually unavoidable. The objective of this work is development of an effective multi scale approximation solution for the Boltzmann equation. Most of the existing methods are based on separation of space and time. The new suggested method is performed without space-time separation. This effective approximation solution is developed on the basis of an expansion for the time derivative of different approximations to the Boltzmann equation. The method of multiple scale expansion is used for the expansion of the time derivative, because the problem of the stiff time behaviour can't be expressed by standard expansion methods. This multiple scale expansion is used in this work to develop approximation solutions for different approximations of the Boltzmann equation, starting from the expansion of the point kinetics equations. The resulting analytic functions are used for testing the applicability and accuracy of the multiple scale expansion method for an approximation solution with 2 delayed neutron groups. The results are tested versus the exact analytical results for the point kinetics equations. Very good agreement between both solutions is obtained. The validity of the solution with 2 delayed neutron groups to approximate the behaviour of the system with 6 delayed neutron groups is demonstrated in an additional analysis. A strategy for a solution with 4 delayed neutron groups is described. A multiple scale expansion is performed for the space-time dependent diffusion equation for one homogenized cell with 2 delayed neutron groups. The result is

Instrument uncertainty predictions

International Nuclear Information System (INIS)

Coutts, D.A.

1991-07-01

The accuracy of measurements and correlations should normally be provided for most experimental activities. The uncertainty is a measure of the accuracy of a stated value or equation. The uncertainty term reflects a combination of instrument errors, modeling limitations, and phenomena understanding deficiencies. This report provides several methodologies to estimate an instrument's uncertainty when used in experimental work. Methods are shown to predict both the pretest and post-test uncertainty

A New Comparison Principle for Impulsive Functional Differential Equations

Directory of Open Access Journals (Sweden)

Gang Li

2015-01-01

Full Text Available We establish a new comparison principle for impulsive differential systems with time delay. Then, using this comparison principle, we obtain some sufficient conditions for several stabilities of impulsive delay differential equations. Finally, we present an example to show the effectiveness of our results.

Optimal control for parabolic-hyperbolic system with time delay

International Nuclear Information System (INIS)

Kowalewski, A.

1985-07-01

In this paper we consider an optimal control problem for a system described by a linear partial differential equation of the parabolic-hyperbolic type with time delay in the state. The right-hand side of this equation and the initial conditions are not continuous functions usually, but they are measurable functions belonging to L 2 or Lsup(infinity) spaces. Therefore, the solution of this equation is given by a certain Sobolev space. The time delay in the state is constant, but it can be also a function of time. The control time T is fixed in our problem. Making use of the Milutin-Dubovicki theorem, necessary and sufficient conditions of optimality with the quadratic performance functional and constrained control are derived for the Dirichlet problem. The flow chart of the algorithm which can be used in the numerical solving of certain optimization problems for distributed systems is also presented. (author)

Characteristic Equation of the Modified Smith predictor to MIMO Systems

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Jorge A. Herrera-Cuartas

2013-11-01

Full Text Available The delay in control systems is a feature frequently in real systems due to the transport of objects or information, a series connection of multiple systems or own processing and sensors delay, among others. Recently there have been several studies to identify the external delay MIMO systems, these works are focused on identification and on-line control of MIMO systems and use a multimodel structure based on modified Smith predictor using different search method. It is clear that for the implementation of the algorithm, and to obtain the convergence and stability analysis, it is necessary to have closed-loop equations of modified Smith predictor. However, in these works is not presented the analytical procedure, not be the main object, displaying only the closed loop equations without the procedure for obtaining it. Therefore, to respond, in this paper, we present an analytical way to derive the closed-loop equations of a modified Smith predictor.  

Delayed feedback control in quantum transport.

Science.gov (United States)

Emary, Clive

2013-09-28

Feedback control in quantum transport has been predicted to give rise to several interesting effects, among them quantum state stabilization and the realization of a mesoscopic Maxwell's daemon. These results were derived under the assumption that control operations on the system are affected instantaneously after the measurement of electronic jumps through it. In this contribution, I describe how to include a delay between detection and control operation in the master equation theory of feedback-controlled quantum transport. I investigate the consequences of delay for the state stabilization and Maxwell's daemon schemes. Furthermore, I describe how delay can be used as a tool to probe coherent oscillations of electrons within a transport system and how this formalism can be used to model finite detector bandwidth.

Periodic and chaotic oscillations in a tumor and immune system interaction model with three delays

International Nuclear Information System (INIS)

Bi, Ping; Ruan, Shigui; Zhang, Xinan

2014-01-01

In this paper, a tumor and immune system interaction model consisted of two differential equations with three time delays is considered in which the delays describe the proliferation of tumor cells, the process of effector cells growth stimulated by tumor cells, and the differentiation of immune effector cells, respectively. Conditions for the asymptotic stability of equilibria and existence of Hopf bifurcations are obtained by analyzing the roots of a second degree exponential polynomial characteristic equation with delay dependent coefficients. It is shown that the positive equilibrium is asymptotically stable if all three delays are less than their corresponding critical values and Hopf bifurcations occur if any one of these delays passes through its critical value. Numerical simulations are carried out to illustrate the rich dynamical behavior of the model with different delay values including the existence of regular and irregular long periodic oscillations

Cucker-Smale model with normalized communication weights and time delay

KAUST Repository

Choi, Young-Pil

2017-03-06

We study a Cucker-Smale-type system with time delay in which agents interact with each other through normalized communication weights. We construct a Lyapunov functional for the system and provide sufficient conditions for asymptotic flocking, i.e., convergence to a common velocity vector. We also carry out a rigorous limit passage to the mean-field limit of the particle system as the number of particles tends to infinity. For the resulting Vlasov-type equation we prove the existence, stability and large-time behavior of measure-valued solutions. This is, to our best knowledge, the first such result for a Vlasov-type equation with time delay. We also present numerical simulations of the discrete system with few particles that provide further insights into the flocking and oscillatory behaviors of the particle velocities depending on the size of the time delay.

Assessing the efficiency of Wolbachia driven Aedes mosquito suppression by delay differential equations.

Science.gov (United States)

Huang, Mugen; Luo, Jiaowan; Hu, Linchao; Zheng, Bo; Yu, Jianshe

2017-12-14

To suppress wild population of Aedes mosquitoes, the primary transmission vector of life-threatening diseases such as dengue, malaria, and Zika, an innovative strategy is to release male mosquitoes carrying the bacterium Wolbachia into natural areas to drive female sterility by cytoplasmic incompatibility. We develop a model of delay differential equations, incorporating the strong density restriction in the larval stage, to assess the delicate impact of life table parameters on suppression efficiency. Through mathematical analysis, we find the sufficient and necessary condition for global stability of the complete suppression state. This condition, combined with the experimental data for Aedes albopictus population in Guangzhou, helps us predict a large range of releasing intensities for suppression success. In particular, we find that if the number of released infected males is no less than four times the number of mosquitoes in wild areas, then the mosquito density in the peak season can be reduced by 95%. We introduce an index to quantify the dependence of suppression efficiency on parameters. The invariance of some quantitative properties of the index values under various perturbations of the same parameter justifies the applicability of this index, and the robustness of our modeling approach. The index yields a ranking of the sensitivity of all parameters, among which the adult mortality has the highest sensitivity and is considerably more sensitive than the natural larvae mortality. Copyright © 2017 Elsevier Ltd. All rights reserved.

Nonlinear wave equation in frequency domain: accurate modeling of ultrafast interaction in anisotropic nonlinear media

DEFF Research Database (Denmark)

Guo, Hairun; Zeng, Xianglong; Zhou, Binbin

2013-01-01

We interpret the purely spectral forward Maxwell equation with up to third-order induced polarizations for pulse propagation and interactions in quadratic nonlinear crystals. The interpreted equation, also named the nonlinear wave equation in the frequency domain, includes quadratic and cubic...... nonlinearities, delayed Raman effects, and anisotropic nonlinearities. The full potential of this wave equation is demonstrated by investigating simulations of solitons generated in the process of ultrafast cascaded second-harmonic generation. We show that a balance in the soliton delay can be achieved due...

Existence of Positive Solutions to a Boundary Value Problem for a Delayed Nonlinear Fractional Differential System

Directory of Open Access Journals (Sweden)

Chen Yuming

2011-01-01

Full Text Available Though boundary value problems for fractional differential equations have been extensively studied, most of the studies focus on scalar equations and the fractional order between 1 and 2. On the other hand, delay is natural in practical systems. However, not much has been done for fractional differential equations with delays. Therefore, in this paper, we consider a boundary value problem of a general delayed nonlinear fractional system. With the help of some fixed point theorems and the properties of the Green function, we establish several sets of sufficient conditions on the existence of positive solutions. The obtained results extend and include some existing ones and are illustrated with some examples for their feasibility.

Singular Perturbation Analysis and Gene Regulatory Networks with Delay

Science.gov (United States)

Shlykova, Irina; Ponosov, Arcady

2009-09-01

There are different ways of how to model gene regulatory networks. Differential equations allow for a detailed description of the network's dynamics and provide an explicit model of the gene concentration changes over time. Production and relative degradation rate functions used in such models depend on the vector of steeply sloped threshold functions which characterize the activity of genes. The most popular example of the threshold functions comes from the Boolean network approach, where the threshold functions are given by step functions. The system of differential equations becomes then piecewise linear. The dynamics of this system can be described very easily between the thresholds, but not in the switching domains. For instance this approach fails to analyze stationary points of the system and to define continuous solutions in the switching domains. These problems were studied in [2], [3], but the proposed model did not take into account a time delay in cellular systems. However, analysis of real gene expression data shows a considerable number of time-delayed interactions suggesting that time delay is essential in gene regulation. Therefore, delays may have a great effect on the dynamics of the system presenting one of the critical factors that should be considered in reconstruction of gene regulatory networks. The goal of this work is to apply the singular perturbation analysis to certain systems with delay and to obtain an analog of Tikhonov's theorem, which provides sufficient conditions for constracting the limit system in the delay case.

Global existence of periodic solutions on a simplified BAM neural network model with delays

International Nuclear Information System (INIS)

Zheng Baodong; Zhang Yazhuo; Zhang Chunrui

2008-01-01

A simplified n-dimensional BAM neural network model with delays is considered. Some results of Hopf bifurcations occurring at the zero equilibrium as the delay increases are exhibited. Global existence of periodic solutions are established using a global Hopf bifurcation result of Wu [Wu J. Symmetric functional-differential equations and neural networks with memory. Trans Am Math Soc 1998;350:4799-838], and a Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney [Li MY, Muldowney J. On Bendixson's criterion. J Differ Equations 1994;106:27-39]. Finally, computer simulations are performed to illustrate the analytical results found

POSITIVE SOLUTIONS TO TWO TYPES OF NEUTRAL DIFFERENTIAL EQUATIONS WITH DISTRIBUTED DELAY

Institute of Scientific and Technical Information of China (English)

无

2012-01-01

In this paper, we study two types of neutral functional differential equations with finite or unbounded distributed deviating arguments. By Banach contraction princi-ple, we obtain some sufficient conditions for the existence of positive solutions to such equations.

Maximal monotone model with delay term of convolution

Directory of Open Access Journals (Sweden)

Claude-Henri Lamarque

2005-01-01

Full Text Available Mechanical models are governed either by partial differential equations with boundary conditions and initial conditions (e.g., in the frame of continuum mechanics or by ordinary differential equations (e.g., after discretization via Galerkin procedure or directly from the model description with the initial conditions. In order to study dynamical behavior of mechanical systems with a finite number of degrees of freedom including nonsmooth terms (e.g., friction, we consider here problems governed by differential inclusions. To describe effects of particular constitutive laws, we add a delay term. In contrast to previous papers, we introduce delay via a Volterra kernel. We provide existence and uniqueness results by using an Euler implicit numerical scheme; then convergence with its order is established. A few numerical examples are given.

Stability analysis of a class of fractional delay differential equations

Indian Academy of Sciences (India)

Abstract. In this paper we analyse stability of nonlinear fractional order delay differential equa- tions of the form Dα y(t) = af (y(t − τ )) − by(t), where Dα is a Caputo fractional derivative of order 0 < α ≤ 1. We describe stability regions using critical curves. To explain the proposed theory, we discuss fractional order logistic ...

Remarks on the inverse scattering transform associated with toda equations

Science.gov (United States)

Ablowitz, Mark J.; Villorroel, J.

The Inverse Scattering Transforms used to solve both the 2+1 Toda equation and a novel reduction, the Toda differential-delay equations are outlined. There are a number of interesting features associated with these systems and the related scattering theory.

A unified approach for impulsive lag synchronization of chaotic systems with time delay

International Nuclear Information System (INIS)

Li Chuandong; Liao Xiaofeng; Zhang Rong

2005-01-01

In this paper, we propose a unified approach for impulsive lag-synchronization of a class of chaotic systems with time delay by employing the stability theory of impulsive delayed differential equations. Three well-known delayed chaotic systems are presented to illustrate our results. Also, the estimates of the stable regions for these systems are given, respectively

Delay time and tunneling transient phenomena

International Nuclear Information System (INIS)

Garcia-Calderon, Gaston; Villavicencio, Jorge

2002-01-01

Analytic solutions to the time-dependent Schroedinger equation for cutoff wave initial conditions are used to investigate the time evolution of the transmitted probability density for tunneling. For a broad range of values of the potential barrier opacity α, we find that the probability density exhibits two evolving structures. One refers to the propagation of a forerunner related to a time domain resonance [Phys. Rev. A 64, 0121907 (2001)], while the other consists of a semiclassical propagating wave front. We find a regime where the forerunners are absent, corresponding to positive time delays, and show that this regime is characterized by opacities α c . The critical opacity α c is derived from the analytical expression for the delay time, which reflects a link between transient effects in tunneling and the delay time

A feedback control model for network flow with multiple pure time delays

Science.gov (United States)

Press, J.

1972-01-01

A control model describing a network flow hindered by multiple pure time (or transport) delays is formulated. Feedbacks connect each desired output with a single control sector situated at the origin. The dynamic formulation invokes the use of differential difference equations. This causes the characteristic equation of the model to consist of transcendental functions instead of a common algebraic polynomial. A general graphical criterion is developed to evaluate the stability of such a problem. A digital computer simulation confirms the validity of such criterion. An optimal decision making process with multiple delays is presented.

Effects of instrumentation time on microleakage of resin-modified glass ionomer cements.

Science.gov (United States)

Yap, Adrian U J; Yeo, Egwin J C; Yap, W Y; Ong, Debbie S B; Tan, Jane W S

2003-01-01

This study investigated the effect of instrumentation time on the microleakage of resin-modified glass ionomer cements (RMGICs). Class V cavities were prepared on buccal and lingual/ palatal surfaces of 64 freshly extracted non-carious premolars. The cavities on each tooth were restored with Fuji II LC (FT [GC]) and Photac-Fil Quick (PF [3M-ESPE]). The restored teeth were randomly divided into two groups of 32 teeth. Finishing/polishing was done immediately after light-polymerization in one group and was delayed for one week in the other group. The following finishing/polishing systems were evaluated: (a) Robot Carbides (RC); (b) SuperSnap (SS); (c) OneGloss (OG) and (d) CompoSite Polishers (CS). The sample size for each instrumentation time, material and finishing/polishing system combination was 8. Storage medium for both immediate and delayed instrumentation groups was distilled water at 37 degrees C during the hiatus period. The teeth were subsequently subjected to dye penetration testing (0.5% basic fushcin), sectioned and scored. Data were analyzed using Kruskal-Wallis and Mann-Whitney U tests at significance level 0.05. For PF, significant difference in enamel leakage was observed between immediate and delayed instrumentation with SS and CS. Significant differences in dentin leakage were also observed between the two instrumentation times for SS. For FT, significant differences in leakage between instrumentation times were observed only in dentin and with RC. Where significant differences in dye penetration scores existed, delayed finishing/polishing resulted in less microleakage.

Stability analysis of impulsive functional differential equations

CERN Document Server

Stamova, Ivanka

2009-01-01

This book is devoted to impulsive functional differential equations which are a natural generalization of impulsive ordinary differential equations (without delay) and of functional differential equations (without impulses). At the present time the qualitative theory of such equationsis under rapid development. After a presentation of the fundamental theory of existence, uniqueness and continuability of solutions, a systematic development of stability theory for that class of problems is given which makes the book unique. It addresses to a wide audience such as mathematicians, applied research

Discrete q-derivatives and symmetries of q-difference equations

Energy Technology Data Exchange (ETDEWEB)

Levi, D [Dipartimento di Fisica, Universita Roma Tre and INFN-Sezione di Roma Tre, Via della Vasca Navale 84, 00146 Rome (Italy); Negro, J [Departamento de FIsica Teorica, Universidad de Valladolid, E-47011, Valladolid (Spain); Olmo, M A del [Departamento de FIsica Teorica, Universidad de Valladolid, E-47011, Valladolid (Spain)

2004-03-12

In this paper we extend the umbral calculus, developed to deal with difference equations on uniform lattices, to q-difference equations. We show that many properties considered for shift invariant difference operators satisfying the umbral calculus can be implemented to the case of the q-difference operators. This q-umbral calculus can be used to provide solutions to linear q-difference equations and q-differential delay equations. To illustrate the method, we will apply the obtained results to the construction of symmetry solutions for the q-heat equation.

Advanced functional evolution equations and inclusions

CERN Document Server

Benchohra, Mouffak

2015-01-01

This book presents up-to-date results on abstract evolution equations and differential inclusions in infinite dimensional spaces. It covers equations with time delay and with impulses, and complements the existing literature in functional differential equations and inclusions. The exposition is devoted to both local and global mild solutions for some classes of functional differential evolution equations and inclusions, and other densely and non-densely defined functional differential equations and inclusions in separable Banach spaces or in Fréchet spaces. The tools used include classical fixed points theorems and the measure-of non-compactness, and each chapter concludes with a section devoted to notes and bibliographical remarks. This monograph is particularly useful for researchers and graduate students studying pure and applied mathematics, engineering, biology and all other applied sciences.

Robotic reactions: Delay-induced patterns in autonomous vehicle systems

Science.gov (United States)

Orosz, Gábor; Moehlis, Jeff; Bullo, Francesco

2010-02-01

Fundamental design principles are presented for vehicle systems governed by autonomous cruise control devices. By analyzing the corresponding delay differential equations, it is shown that for any car-following model short-wavelength oscillations can appear due to robotic reaction times, and that there are tradeoffs between the time delay and the control gains. The analytical findings are demonstrated on an optimal velocity model using numerical continuation and numerical simulation.

Robotic reactions: delay-induced patterns in autonomous vehicle systems.

Science.gov (United States)

Orosz, Gábor; Moehlis, Jeff; Bullo, Francesco

2010-02-01

Fundamental design principles are presented for vehicle systems governed by autonomous cruise control devices. By analyzing the corresponding delay differential equations, it is shown that for any car-following model short-wavelength oscillations can appear due to robotic reaction times, and that there are tradeoffs between the time delay and the control gains. The analytical findings are demonstrated on an optimal velocity model using numerical continuation and numerical simulation.

Analytical and experimental study of two delay-coupled excitable units.

Science.gov (United States)

Weicker, Lionel; Erneux, Thomas; Keuninckx, Lars; Danckaert, Jan

2014-01-01

We investigate the onset of time-periodic oscillations for a system of two identical delay-coupled excitable (nonoscillatory) units. We first analyze these solutions by using asymptotic methods. The oscillations are described as relaxation oscillations exhibiting successive slow and fast changes. The analysis highlights the determinant role of the delay during the fast transition layers. We then study experimentally a system of two coupled electronic circuits that is modeled mathematically by the same delay differential equations. We obtain quantitative agreements between analytical and experimental bifurcation diagrams.

A delay-dependent approach to robust control for neutral uncertain neural networks with mixed interval time-varying delays

International Nuclear Information System (INIS)

Lu, Chien-Yu

2011-01-01

This paper considers the problem of delay-dependent global robust stabilization for discrete, distributed and neutral interval time-varying delayed neural networks described by nonlinear delay differential equations of the neutral type. The parameter uncertainties are norm bounded. The activation functions are assumed to be bounded and globally Lipschitz continuous. Using a Lyapunov functional approach and linear matrix inequality (LMI) techniques, the stability criteria for the uncertain neutral neural networks with interval time-varying delays are established in the form of LMIs, which can be readily verified using the standard numerical software. An important feature of the result reported is that all the stability conditions are dependent on the upper and lower bounds of the delays. Another feature of the results lies in that it involves fewer free weighting matrix strategy, and upper bounds of the inner product between two vectors are not introduced to reduce the conservatism of the criteria. Two illustrative examples are provided to demonstrate the effectiveness and the reduced conservatism of the proposed method

Stability of Vector Functional Differential Equations: A Survey | Gil ...

African Journals Online (AJOL)

This paper is a survey of the recent results of the author on the stability of linear and nonlinear vector differential equations with delay. Explicit conditions for the exponential and absolute stabilities are derived. Moreover, solution estimates for the considered equations are established. They provide the bounds for the regions ...

The solutions of second-order linear differential systems with constant delays

Science.gov (United States)

Diblík, Josef; Svoboda, Zdeněk

2017-07-01

The representations of solutions to initial problems for non-homogenous n-dimensional second-order differential equations with delays x″(t )-2 A x'(t -τ )+(A2+B2)x (t -2 τ )=f (t ) by means of special matrix delayed functions are derived. Square matrices A and B are commuting and τ > 0. Derived representations use what is called a delayed exponential of a matrix and results generalize some of known results previously derived for homogenous systems.

Charging-delay induced dust acoustic collisionless shock wave: Roles of negative ions

International Nuclear Information System (INIS)

Ghosh, Samiran; Bharuthram, R.; Khan, Manoranjan; Gupta, M. R.

2006-01-01

The effects of charging-delay and negative ions on nonlinear dust acoustic waves are investigated. It has been found that the charging-delay induced anomalous dissipation causes generation of dust acoustic collisionless shock waves in an electronegative dusty plasma. The small but finite amplitude wave is governed by a Korteweg-de Vries Burger equation in which the Burger term arises due to the charging-delay. Numerical investigations reveal that the charging-delay induced dissipation and shock strength decreases (increases) with the increase of negative ion concentration (temperature)

Numerical solution of the point reactor kinetics equations with fuel burn-up and temperature feedback

International Nuclear Information System (INIS)

Tashakor, S.; Jahanfarnia, G.; Hashemi-Tilehnoee, M.

2010-01-01

Point reactor kinetics equations are solved numerically using one group of delayed neutrons and with fuel burn-up and temperature feedback included. To calculate the fraction of one-group delayed neutrons, a group of differential equations are solved by an implicit time method. Using point reactor kinetics equations, changes in mean neutrons density, temperature, and reactivity are calculated in different times during the reactor operation. The variation of reactivity, temperature, and maximum power with time are compared with the predictions by other methods.

Time-delayed autosynchronous swarm control.

Science.gov (United States)

Biggs, James D; Bennet, Derek J; Dadzie, S Kokou

2012-01-01

In this paper a general Morse potential model of self-propelling particles is considered in the presence of a time-delayed term and a spring potential. It is shown that the emergent swarm behavior is dependent on the delay term and weights of the time-delayed function, which can be set to induce a stationary swarm, a rotating swarm with uniform translation, and a rotating swarm with a stationary center of mass. An analysis of the mean field equations shows that without a spring potential the motion of the center of mass is determined explicitly by a multivalued function. For a nonzero spring potential the swarm converges to a vortex formation about a stationary center of mass, except at discrete bifurcation points where the center of mass will periodically trace an ellipse. The analytical results defining the behavior of the center of mass are shown to correspond with the numerical swarm simulations.

High-explosive-driven delay line pulse generator

International Nuclear Information System (INIS)

Shearer, J.W.

1982-01-01

The inclusion of a delay line circuit into the design of a high-explosive-driven generator shortens the time constant of the output pulse. After a brief review of generator concepts and previously described pulse-shortening methods, a geometry is presented which incorporates delay line circuit techcniques into a coil generator. The circuit constants are adjusted to match the velocity of the generated electromagnetic wave to the detonation velocity of the high explosive. The proposed generator can be modeled by adding a variable inductance term to the telegrapher's equation. A particular solution of this equation is useful for exploring the operational parameters of the generator. The duration of the electromagnetic pulse equals the radial expansion time of the high-explosive-driven armature until it strikes the coil. Because the impedance of the generator is a constant, the current multiplication factor is limited only by nonlinear effects such as voltage breakdown, diffusion, and compression at high energies

Transition among synchronous schemes in coupled nonidentical multiple time delay systems

International Nuclear Information System (INIS)

Thang Manh Hoang

2009-01-01

We present the transition among possible synchronous schemes in coupled nonidentical multiple time delay systems, i.e., lag, projective-lag, complete, anticipating and projective-anticipating synchronization. The number of nonlinear transforms in the master's equation can be different from that in slave's, and nonlinear transforms can be in various forms. The driving signal is the sum of nonlinearly transformed components of delayed state variable. Moreover, the equation representing for driving signal is constructed exactly so that the difference between the master's and slave's structures is complemented. The sufficient condition for synchronization is considered by the Krasovskii-Lyapunov theory. The specific examples will demonstrate and verify the effectiveness of the proposed models.

A Generalized Halanay Inequality for Stability of Nonlinear Neutral Functional Differential Equations

Directory of Open Access Journals (Sweden)

Wansheng Wang

2010-01-01

Full Text Available This paper is devoted to generalize Halanay's inequality which plays an important rule in study of stability of differential equations. By applying the generalized Halanay inequality, the stability results of nonlinear neutral functional differential equations (NFDEs and nonlinear neutral delay integrodifferential equations (NDIDEs are obtained.

Periodicity computation of generalized mathematical biology problems involving delay differential equations.

Science.gov (United States)

Jasim Mohammed, M; Ibrahim, Rabha W; Ahmad, M Z

2017-03-01

In this paper, we consider a low initial population model. Our aim is to study the periodicity computation of this model by using neutral differential equations, which are recognized in various studies including biology. We generalize the neutral Rayleigh equation for the third-order by exploiting the model of fractional calculus, in particular the Riemann-Liouville differential operator. We establish the existence and uniqueness of a periodic computational outcome. The technique depends on the continuation theorem of the coincidence degree theory. Besides, an example is presented to demonstrate the finding.

Global Hopf bifurcation analysis on a BAM neural network with delays

Science.gov (United States)

Sun, Chengjun; Han, Maoan; Pang, Xiaoming

2007-01-01

A delayed differential equation that models a bidirectional associative memory (BAM) neural network with four neurons is considered. By using a global Hopf bifurcation theorem for FDE and a Bendixon's criterion for high-dimensional ODE, a group of sufficient conditions for the system to have multiple periodic solutions are obtained when the sum of delays is sufficiently large.

Global Hopf bifurcation analysis on a BAM neural network with delays

International Nuclear Information System (INIS)

Sun Chengjun; Han Maoan; Pang Xiaoming

2007-01-01

A delayed differential equation that models a bidirectional associative memory (BAM) neural network with four neurons is considered. By using a global Hopf bifurcation theorem for FDE and a Bendixon's criterion for high-dimensional ODE, a group of sufficient conditions for the system to have multiple periodic solutions are obtained when the sum of delays is sufficiently large

RECOVERY OF LARGE ANGULAR SCALE CMB POLARIZATION FOR INSTRUMENTS EMPLOYING VARIABLE-DELAY POLARIZATION MODULATORS

Energy Technology Data Exchange (ETDEWEB)

Miller, N. J.; Marriage, T. A.; Appel, J. W.; Bennett, C. L.; Eimer, J.; Essinger-Hileman, T.; Harrington, K.; Rostem, K.; Watts, D. J. [Department of Physics and Astronomy, Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218 (United States); Chuss, D. T. [Department of Physics, Villanova University, 800 E Lancaster, Villanova, PA 19085 (United States); Wollack, E. J.; Fixsen, D. J.; Moseley, S. H.; Switzer, E. R., E-mail: Nathan.J.Miller@nasa.gov [Observational Cosmology Laboratory, Code 665, NASA Goddard Space Flight Center, Greenbelt, MD 20771 (United States)

2016-02-20

Variable-delay Polarization Modulators (VPMs) are currently being implemented in experiments designed to measure the polarization of the cosmic microwave background on large angular scales because of their capability for providing rapid, front-end polarization modulation and control over systematic errors. Despite the advantages provided by the VPM, it is important to identify and mitigate any time-varying effects that leak into the synchronously modulated component of the signal. In this paper, the effect of emission from a 300 K VPM on the system performance is considered and addressed. Though instrument design can greatly reduce the influence of modulated VPM emission, some residual modulated signal is expected. VPM emission is treated in the presence of rotational misalignments and temperature variation. Simulations of time-ordered data are used to evaluate the effect of these residual errors on the power spectrum. The analysis and modeling in this paper guides experimentalists on the critical aspects of observations using VPMs as front-end modulators. By implementing the characterizations and controls as described, front-end VPM modulation can be very powerful for mitigating 1/f noise in large angular scale polarimetric surveys. None of the systematic errors studied fundamentally limit the detection and characterization of B-modes on large scales for a tensor-to-scalar ratio of r = 0.01. Indeed, r < 0.01 is achievable with commensurately improved characterizations and controls.

Oscillatory dynamics of an intravenous glucose tolerance test model with delay interval

Science.gov (United States)

Shi, Xiangyun; Kuang, Yang; Makroglou, Athena; Mokshagundam, Sriprakash; Li, Jiaxu

2017-11-01

Type 2 diabetes mellitus (T2DM) has become prevalent pandemic disease in view of the modern life style. Both diabetic population and health expenses grow rapidly according to American Diabetes Association. Detecting the potential onset of T2DM is an essential focal point in the research of diabetes mellitus. The intravenous glucose tolerance test (IVGTT) is an effective protocol to determine the insulin sensitivity, glucose effectiveness, and pancreatic β-cell functionality, through the analysis and parameter estimation of a proper differential equation model. Delay differential equations have been used to study the complex physiological phenomena including the glucose and insulin regulations. In this paper, we propose a novel approach to model the time delay in IVGTT modeling. This novel approach uses two parameters to simulate not only both discrete time delay and distributed time delay in the past interval, but also the time delay distributed in a past sub-interval. Normally, larger time delay, either a discrete or a distributed delay, will destabilize the system. However, we find that time delay over a sub-interval might not. We present analytically some basic model properties, which are desirable biologically and mathematically. We show that this relatively simple model provides good fit to fluctuating patient data sets and reveals some intriguing dynamics. Moreover, our numerical simulation results indicate that our model may remove the defect in well known Minimal Model, which often overestimates the glucose effectiveness index.

Bifurcation analysis and spatio-temporal patterns of nonlinear oscillations in a delayed neural network with unidirectional coupling

International Nuclear Information System (INIS)

Song Yongli; Tadé, Moses O; Zhang Tonghua

2009-01-01

In this paper, a delayed neural network with unidirectional coupling is considered which consists of two two-dimensional nonlinear differential equation systems with exponential decay where one system receives a delayed input from the other system. Some parameter regions are given for conditional/absolute stability and Hopf bifurcations by using the theory of functional differential equations. Conditions ensuring the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the centre manifold theorem. We also investigate the spatio-temporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay-differential equations combined with representation theory of Lie groups. Then the global continuation of phase-locked periodic solutions is investigated. Numerical simulations are given to illustrate the results obtained

Charging-delay effect on longitudinal dust acoustic shock wave in strongly coupled dusty plasma

International Nuclear Information System (INIS)

Ghosh, Samiran; Gupta, M.R.

2005-01-01

Taking into account the charging-delay effect, the nonlinear propagation characteristics of longitudinal dust acoustic wave in strongly coupled collisional dusty plasma described by generalized hydrodynamic model have been investigated. In the 'hydrodynamic limit', a Korteweg-de Vries Burger (KdVB) equation with a damping term arising due to dust-neutral collision is derived in which the Burger term is proportional to the dissipation due to dust viscosity through dust-dust correlation and charging-delay-induced anomalous dissipation. On the other hand, in the 'kinetic limit', a KdVB equation with a damping term and a nonlocal nonlinear forcing term arising due to memory-dependent strong correlation effect of dust fluid is derived in which the Burger term depends only on the charging-delay-induced dissipation. Numerical solution of integrodifferential equations reveals that (i) dissipation due to dust viscosity and principally due to charging delay causes excitation of the longitudinal dust acoustic shock wave in strongly coupled dusty plasma and (ii) dust-neutral collision does not appear to play any direct role in shock formation. The condition for the generation of shock is also discussed briefly

A simple delay model for two-phase flow dynamics

Energy Technology Data Exchange (ETDEWEB)

Clausse, A.; Delmastro, D.F.; Juanico`, L.E. [Centro Atomico Bariloche (Argentina)

1995-09-01

A model based in delay equations for density-wave oscillations is presented. High Froude numbers and moderate ones were considered. The equations were numerically analyzed and compared with more sophisticated models. The influence of the gravity term was studied. Different kinds of behavior were found, particularly sub-critical and super-critical Hopf bifurcations. Moreover the present approach can be used to better understand the complicated dynamics of boiling flows systems.

Dissolvable fluidic time delays for programming multi-step assays in instrument-free paper diagnostics.

Science.gov (United States)

Lutz, Barry; Liang, Tinny; Fu, Elain; Ramachandran, Sujatha; Kauffman, Peter; Yager, Paul

2013-07-21

Lateral flow tests (LFTs) are an ingenious format for rapid and easy-to-use diagnostics, but they are fundamentally limited to assay chemistries that can be reduced to a single chemical step. In contrast, most laboratory diagnostic assays rely on multiple timed steps carried out by a human or a machine. Here, we use dissolvable sugar applied to paper to create programmable flow delays and present a paper network topology that uses these time delays to program automated multi-step fluidic protocols. Solutions of sucrose at different concentrations (10-70% of saturation) were added to paper strips and dried to create fluidic time delays spanning minutes to nearly an hour. A simple folding card format employing sugar delays was shown to automate a four-step fluidic process initiated by a single user activation step (folding the card); this device was used to perform a signal-amplified sandwich immunoassay for a diagnostic biomarker for malaria. The cards are capable of automating multi-step assay protocols normally used in laboratories, but in a rapid, low-cost, and easy-to-use format.

Delayed neutron fraction and prompt decay constant measurement in the MINERVE reactor using the PSI instrumentation

Energy Technology Data Exchange (ETDEWEB)

Perret, Gregory [Paul Scherrer Institute, Villigen, 5232, (Switzerland)

2015-07-01

The critical decay constant (B/A), delayed neutron fraction (B) and generation time (A) of the Minerve reactor were measured by the Paul Scherrer Institut (PSI) and the Commissariat a l'Energie Atomique (CEA) in September 2014 using the Feynman-alpha and Power Spectral Density neutron noise measurement techniques. Three slightly subcritical configuration were measured using two 1-g {sup 235}U fission chambers. This paper reports on the results obtained by PSI in the near critical configuration (-2g). The most reliable and precise results were obtained with the Cross-Power Spectral Density technique: B = 708.4±9.2 pcm, B/A = 79.0±0.6 s{sup -1} and A 89.7±1.4 micros. Predictions of the same kinetic parameters were obtained with MCNP5-v1.6 and the JEFF-3.1 and ENDF/B-VII.1 nuclear data libraries. On average the predictions for B and B/A overestimate the experimental results by 5% and 11%, respectively. The discrepancy is suspected to come from either a corruption of the data or from the inadequacy of the point kinetic equations to interpret the measurements in the Minerve driven system. (authors)

Learning monopolies with delayed feedback on price expectations

Science.gov (United States)

Matsumoto, Akio; Szidarovszky, Ferenc

2015-11-01

We call the intercept of the price function with the vertical axis the maximum price and the slope of the price function the marginal price. In this paper it is assumed that a monopolistic firm has full information about the marginal price and its own cost function but is uncertain on the maximum price. However, by repeated interaction with the market, the obtained price observations give a basis for an adaptive learning process of the maximum price. It is also assumed that the price observations have fixed delays, so the learning process can be described by a delayed differential equation. In the cases of one or two delays, the asymptotic behavior of the resulting dynamic process is examined, stability conditions are derived. Three main results are demonstrated in the two delay learning processes. First, it is possible to stabilize the equilibrium which is unstable in the one delay model. Second, complex dynamics involving chaos, which is impossible in the one delay model, can emerge. Third, alternations of stability and instability (i.e., stability switches) occur repeatedly.

Analysis of the effects of time delay in clock recovery circuits based on Phase-locked loops

DEFF Research Database (Denmark)

Zibar, Darko; Oxenløwe, Leif Katsuo; Clausen, Anders

2004-01-01

Influence of time delay in a balanced optical phase-locked loops (OPLL) with a proportional integrator (Pl) filter is investigated using a delayed differential equation (DDE) is investigated. The limitations, which a time delay imposes on the Pl filter bandwidth, at increasing values of loop gain...

How the 2SLS/IV estimator can handle equality constraints in structural equation models: a system-of-equations approach.

Science.gov (United States)

Nestler, Steffen

2014-05-01

Parameters in structural equation models are typically estimated using the maximum likelihood (ML) approach. Bollen (1996) proposed an alternative non-iterative, equation-by-equation estimator that uses instrumental variables. Although this two-stage least squares/instrumental variables (2SLS/IV) estimator has good statistical properties, one problem with its application is that parameter equality constraints cannot be imposed. This paper presents a mathematical solution to this problem that is based on an extension of the 2SLS/IV approach to a system of equations. We present an example in which our approach was used to examine strong longitudinal measurement invariance. We also investigated the new approach in a simulation study that compared it with ML in the examination of the equality of two latent regression coefficients and strong measurement invariance. Overall, the results show that the suggested approach is a useful extension of the original 2SLS/IV estimator and allows for the effective handling of equality constraints in structural equation models. © 2013 The British Psychological Society.

Time-dependent solutions for stochastic systems with delays: Perturbation theory and applications to financial physics

International Nuclear Information System (INIS)

Frank, T.D.

2006-01-01

First-order approximations of time-dependent solutions are determined for stochastic systems perturbed by time-delayed feedback forces. To this end, the theory of delay Fokker-Planck equations is applied in combination with Bayes' theorem. Applications to a time-delayed Ornstein-Uhlenbeck process and the geometric Brownian walk of financial physics are discussed

Control-based method to identify underlying delays of a nonlinear dynamical system.

Science.gov (United States)

Yu, Dongchuan; Frasca, Mattia; Liu, Fang

2008-10-01

We suggest several stationary state control-based delay identification methods which do not require any structural information about the controlled systems and are applicable to systems described by delayed ordinary differential equations. This proposed technique includes three steps: (i) driving a system to a steady state; (ii) perturbing the control signal for shifting the steady state; and (iii) identifying all delays by detecting the time that the system is abruptly drawn out of stationarity. Some aspects especially important for applications are discussed as well, including interaction delay identification, stationary state convergence speed, performance comparison, and the influence of noise on delay identification. Several examples are presented to illustrate the reliability and robustness of all delay identification methods suggested.

Multipulse dynamics of a passively mode-locked semiconductor laser with delayed optical feedback

Science.gov (United States)

Jaurigue, Lina; Krauskopf, Bernd; Lüdge, Kathy

2017-11-01

Passively mode-locked semiconductor lasers are compact, inexpensive sources of short light pulses of high repetition rates. In this work, we investigate the dynamics and bifurcations arising in such a device under the influence of time delayed optical feedback. This laser system is modelled by a system of delay differential equations, which includes delay terms associated with the laser cavity and feedback loop. We make use of specialised path continuation software for delay differential equations to analyse the regime of short feedback delays. Specifically, we consider how the dynamics and bifurcations depend on the pump current of the laser, the feedback strength, and the feedback delay time. We show that an important role is played by resonances between the mode-locking frequencies and the feedback delay time. We find feedback-induced harmonic mode locking and show that a mismatch between the fundamental frequency of the laser and that of the feedback cavity can lead to multi-pulse or quasiperiodic dynamics. The quasiperiodic dynamics exhibit a slow modulation, on the time scale of the gain recovery rate, which results from a beating with the frequency introduced in the associated torus bifurcations and leads to gain competition between multiple pulse trains within the laser cavity. Our results also have implications for the case of large feedback delay times, where a complete bifurcation analysis is not practical. Namely, for increasing delay, there is an ever-increasing degree of multistability between mode-locked solutions due to the frequency pulling effect.

Complex Time-Delay Systems Theory and Applications

CERN Document Server

Atay, Fatihcan M

2010-01-01

Time delays in dynamical systems arise as an inevitable consequence of finite speeds of information transmission. Realistic models increasingly demand the inclusion of delays in order to properly understand, analyze, design, and control real-life systems. The goal of this book is to present the state-of-the-art in research on time-delay dynamics in the framework of complex systems and networks. While the mathematical theory of delay equations is quite mature, its application to the particular problems of complex systems and complexity is a newly emerging field, and the present volume aims to play a pioneering role in this perspective. The chapters in this volume are authored by renowned experts and cover both theory and applications in a wide range of fields, with examples extending from neuroscience and biology to laser physics and vehicle traffic. Furthermore, all chapters include sufficient introductory material and extensive bibliographies, making the book a self-contained reference for both students and ...

Stability Switches, Hopf Bifurcations, and Spatio-temporal Patterns in a Delayed Neural Model with Bidirectional Coupling

Science.gov (United States)

Song, Yongli; Zhang, Tonghua; Tadé, Moses O.

2009-12-01

The dynamical behavior of a delayed neural network with bi-directional coupling is investigated by taking the delay as the bifurcating parameter. Some parameter regions are given for conditional/absolute stability and Hopf bifurcations by using the theory of functional differential equations. As the propagation time delay in the coupling varies, stability switches for the trivial solution are found. Conditions ensuring the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. We also discuss the spatio-temporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. In particular, we obtain that the spatio-temporal patterns of bifurcating periodic oscillations will alternate according to the change of the propagation time delay in the coupling, i.e., different ranges of delays correspond to different patterns of neural activities. Numerical simulations are given to illustrate the obtained results and show the existence of bursts in some interval of the time for large enough delay.

Necessary and sufficient conditions for nonoscillatory solutions of impulsive delay differential equations

Directory of Open Access Journals (Sweden)

Shao Yuan Huang

2013-07-01

Full Text Available Monotonicity of solutions is an important property in the investigation of oscillatory behaviors of differential equations. A number of papers provide some existence criteria for eventually positive increasing solutions. However, relatively little attention is paid to eventually positive solutions that are also eventually decreasing solutions. For this reason, we establish several new and sharp oscillatory criteria for impulsive functional differential equations from this viewpoint.

Effect of state-dependent delay on a weakly damped nonlinear oscillator.

Science.gov (United States)

Mitchell, Jonathan L; Carr, Thomas W

2011-04-01

We consider a weakly damped nonlinear oscillator with state-dependent delay, which has applications in models for lasers, epidemics, and microparasites. More generally, the delay-differential equations considered are a predator-prey system where the delayed term is linear and represents the proliferation of the predator. We determine the critical value of the delay that causes the steady state to become unstable to periodic oscillations via a Hopf bifurcation. Using asymptotic averaging, we determine how the system's behavior is influenced by the functional form of the state-dependent delay. Specifically, we determine whether the branch of periodic solutions will be either sub- or supercritical as well as an accurate estimation of the amplitude. Finally, we choose a few examples of state-dependent delay to test our analytical results by comparing them to numerical continuation.

Stability and Hopf Bifurcation of a Reaction-Diffusion Neutral Neuron System with Time Delay

Science.gov (United States)

Dong, Tao; Xia, Linmao

2017-12-01

In this paper, a type of reaction-diffusion neutral neuron system with time delay under homogeneous Neumann boundary conditions is considered. By constructing a basis of phase space based on the eigenvectors of the corresponding Laplace operator, the characteristic equation of this system is obtained. Then, by selecting time delay and self-feedback strength as the bifurcating parameters respectively, the dynamic behaviors including local stability and Hopf bifurcation near the zero equilibrium point are investigated when the time delay and self-feedback strength vary. Furthermore, the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are obtained by using the normal form and the center manifold theorem for the corresponding partial differential equation. Finally, two simulation examples are given to verify the theory.

Nonlinear free vibration control of beams using acceleration delayed-feedback control

International Nuclear Information System (INIS)

Alhazza, Khaled A; Alajmi, Mohammed; Masoud, Ziyad N

2008-01-01

A single-mode delayed-feedback control strategy is developed to reduce the free vibrations of a flexible beam using a piezoelectric actuator. A nonlinear variational model of the beam based on the von Kàrmàn nonlinear type deformations is considered. Using Galerkin's method, the resulting governing partial differential equations of motion are reduced to a system of nonlinear ordinary differential equations. A linear model using the first mode is derived and is used to characterize the damping produced by the controller as a function of the controller's gain and delay. Three-dimensional figures showing the damping magnitude as a function of the controller gain and delay are presented. The characteristic damping of the controller as predicted by the linear model is compared to that calculated using direct long-time integration of a three-mode nonlinear model. Optimal values of the controller gain and delay using both methods are obtained, simulated and compared. To validate the single-mode approximation, numerical simulations are performed using a three-mode full nonlinear model. Results of the simulations demonstrate an excellent controller performance in mitigating the first-mode vibration

Finite-Dimensional Representations for Controlled Diffusions with Delay

Energy Technology Data Exchange (ETDEWEB)

Federico, Salvatore, E-mail: salvatore.federico@unimi.it [Università di Milano, Dipartimento di Economia, Management e Metodi Quantitativi (Italy); Tankov, Peter, E-mail: tankov@math.univ-paris-diderot.fr [Université Paris Diderot, Laboratoire de Probabilités et Modèles Aléatoires (France)

2015-02-15

We study stochastic delay differential equations (SDDE) where the coefficients depend on the moving averages of the state process. As a first contribution, we provide sufficient conditions under which the solution of the SDDE and a linear path functional of it admit a finite-dimensional Markovian representation. As a second contribution, we show how approximate finite-dimensional Markovian representations may be constructed when these conditions are not satisfied, and provide an estimate of the error corresponding to these approximations. These results are applied to optimal control and optimal stopping problems for stochastic systems with delay.

Fission delay and GDR γ-ray from very heavy system

International Nuclear Information System (INIS)

Shen, W.Q.; Wang, J.S.; Ye, W.; Cai, Y.H.; Ma, Y.G.; Feng, J.; Fang, D.Q.; Cai, X.Z.

1999-01-01

The study of the fission delay in reaction of 84 Kr+ 27 Al at 10.6 Mev/u and the systematics of fission delay are described. Authors also discussed the possibility to study the GDR γ rays emitted from the super-heavy compound system on the basis of the strong increasing of the GDR γ rays duo to the fission delay. The calculation results of the GDR γ rays from the super-heavy compound system via microscopic semi-classical Vlasov equation and the experimental data analysis for searching the super-heavy compound system via GDR γ were given

A normalized PID controller in networked control systems with varying time delays.

Science.gov (United States)

Tran, Hoang-Dung; Guan, Zhi-Hong; Dang, Xuan-Kien; Cheng, Xin-Ming; Yuan, Fu-Shun

2013-09-01

It requires not only simplicity and flexibility but also high specified stability and robustness of system to design a PI/PID controller in such complicated networked control systems (NCSs) with delays. By gain and phase margins approach, this paper proposes a novel normalized PI/PID controller for NCSs based on analyzing the stability and robustness of system under the effect of network-induced delays. Specifically, We take into account the total measured network delays to formulate the gain and phase margins of the closed-loop system in the form of a set of equations. With pre-specified values of gain and phase margins, this set of equations is then solved for calculating the closed forms of control parameters which enable us to propose the normalized PI/PID controller simultaneously satisfying the following two requirements: (1) simplicity without re-solving the optimization problem for a new process, (2) high flexibility to cope with large scale of random delays and deal with many different processes in different conditions of network. Furthermore, in our method, the upper bound of random delay can be estimated to indicate the operating domain of proposed PI/PID controller. Finally, simulation results are shown to demonstrate the advantages of our proposed controller in many situations of network-induced delays. Copyright © 2013 ISA. Published by Elsevier Ltd. All rights reserved.

Phase models and clustering in networks of oscillators with delayed coupling

Science.gov (United States)

Campbell, Sue Ann; Wang, Zhen

2018-01-01

We consider a general model for a network of oscillators with time delayed coupling where the coupling matrix is circulant. We use the theory of weakly coupled oscillators to reduce the system of delay differential equations to a phase model where the time delay enters as a phase shift. We use the phase model to determine model independent existence and stability results for symmetric cluster solutions. Our results extend previous work to systems with time delay and a more general coupling matrix. We show that the presence of the time delay can lead to the coexistence of multiple stable clustering solutions. We apply our analytical results to a network of Morris Lecar neurons and compare these results with numerical continuation and simulation studies.

A simple model of carcinogenic mutations with time delay and diffusion.

Science.gov (United States)

Piotrowska, Monika Joanna; Foryś, Urszula; Bodnar, Marek; Poleszczuk, Jan

2013-06-01

In the paper we consider a system of delay differential equations (DDEs) of Lotka-Volterra type with diffusion reflecting mutations from normal to malignant cells. The model essentially follows the idea of Ahangar and Lin (2003) where mutations in three different environmental conditions, namely favorable, competitive and unfavorable, were considered. We focus on the unfavorable conditions that can result from a given treatment, e.g. chemotherapy. Included delay stands for the interactions between benign and other cells. We compare the dynamics of ODEs system, the system with delay and the system with delay and diffusion. We mainly focus on the dynamics when a positive steady state exists. The system which is globally stable in the case without the delay and diffusion is destabilized by increasing delay, and therefore the underlying kinetic dynamics becomes oscillatory due to a Hopf bifurcation for appropriate values of the delay. This suggests the occurrence of spatially non-homogeneous periodic solutions for the system with the delay and diffusion.

A ternary logic model for recurrent neuromime networks with delay.

Science.gov (United States)

Hangartner, R D; Cull, P

1995-07-01

In contrast to popular recurrent artificial neural network (RANN) models, biological neural networks have unsymmetric structures and incorporate significant delays as a result of axonal propagation. Consequently, biologically inspired neural network models are more accurately described by nonlinear differential-delay equations rather than nonlinear ordinary differential equations (ODEs), and the standard techniques for studying the dynamics of RANNs are wholly inadequate for these models. This paper develops a ternary-logic based method for analyzing these networks. Key to the technique is the realization that a nonzero delay produces a bounded stability region. This result significantly simplifies the construction of sufficient conditions for characterizing the network equilibria. If the network gain is large enough, each equilibrium can be classified as either asymptotically stable or unstable. To illustrate the analysis technique, the swim central pattern generator (CPG) of the sea slug Tritonia diomedea is examined. For wide range of reasonable parameter values, the ternary analysis shows that none of the network equilibria are stable, and thus the network must oscillate. The results show that complex synaptic dynamics are not necessary for pattern generation.

A Cucker--Smale Model with Noise and Delay

KAUST Repository

Erban, Radek

2016-08-09

A generalization of the Cucker-Smale model for collective animal behavior is investigated. The model is formulated as a system of delayed stochastic differential equations. It incorporates two additional processes which are present in animal decision making, but are often neglected in modeling: (i) stochasticity (imperfections) of individual behavior and (ii) delayed responses of individuals to signals in their environment. Sufficient conditions for flocking for the generalized Cucker-Smale model are derived by using a suitable Lyapunov functional. As a by-product, a new result regarding the asymptotic behavior of delayed geometric Brownian motion is obtained. In the second part of the paper, results of systematic numerical simulations are presented. They not only illustrate the analytical results, but hint at a somehow surprising behavior

A Cucker--Smale Model with Noise and Delay

KAUST Repository

Erban, Radek; Haskovec, Jan; Sun, Yongzheng

2016-01-01

A generalization of the Cucker-Smale model for collective animal behavior is investigated. The model is formulated as a system of delayed stochastic differential equations. It incorporates two additional processes which are present in animal decision making, but are often neglected in modeling: (i) stochasticity (imperfections) of individual behavior and (ii) delayed responses of individuals to signals in their environment. Sufficient conditions for flocking for the generalized Cucker-Smale model are derived by using a suitable Lyapunov functional. As a by-product, a new result regarding the asymptotic behavior of delayed geometric Brownian motion is obtained. In the second part of the paper, results of systematic numerical simulations are presented. They not only illustrate the analytical results, but hint at a somehow surprising behavior

Efficiently and easily integrating differential equations with JiTCODE, JiTCDDE, and JiTCSDE

Science.gov (United States)

Ansmann, Gerrit

2018-04-01

We present a family of Python modules for the numerical integration of ordinary, delay, or stochastic differential equations. The key features are that the user enters the derivative symbolically and it is just-in-time-compiled, allowing the user to efficiently integrate differential equations from a higher-level interpreted language. The presented modules are particularly suited for large systems of differential equations such as those used to describe dynamics on complex networks. Through the selected method of input, the presented modules also allow almost complete automatization of the process of estimating regular as well as transversal Lyapunov exponents for ordinary and delay differential equations. We conceptually discuss the modules' design, analyze their performance, and demonstrate their capabilities by application to timely problems.

Asymptotic Behaviour and Extinction of Delay Lotka-Volterra Model with Jump-Diffusion

Directory of Open Access Journals (Sweden)

Dan Li

2014-01-01

Full Text Available This paper studies the effect of jump-diffusion random environmental perturbations on the asymptotic behaviour and extinction of Lotka-Volterra population dynamics with delays. The contributions of this paper lie in the following: (a to consider delay stochastic differential equation with jumps, we introduce a proper initial data space, in which the initial data may be discontinuous function with downward jumps; (b we show that the delay stochastic differential equation with jumps associated with our model has a unique global positive solution and give sufficient conditions that ensure stochastically ultimate boundedness, moment average boundedness in time, and asymptotic polynomial growth of our model; (c the sufficient conditions for the extinction of the system are obtained, which generalized the former results and showed that the sufficiently large random jump magnitudes and intensity (average rate of jump events arrival may lead to extinction of the population.

Power-spectral-density relationship for retarded differential equations

Science.gov (United States)

Barker, L. K.

1974-01-01

The power spectral density (PSD) relationship between input and output of a set of linear differential-difference equations of the retarded type with real constant coefficients and delays is discussed. The form of the PSD relationship is identical with that applicable to unretarded equations. Since the PSD relationship is useful if and only if the system described by the equations is stable, the stability must be determined before applying the PSD relationship. Since it is sometimes difficult to determine the stability of retarded equations, such equations are often approximated by simpler forms. It is pointed out that some common approximations can lead to erroneous conclusions regarding the stability of a system and, therefore, to the possibility of obtaining PSD results which are not valid.

Taylor's series method for solving the nonlinear point kinetics equations

International Nuclear Information System (INIS)

Nahla, Abdallah A.

2011-01-01

Highlights: → Taylor's series method for nonlinear point kinetics equations is applied. → The general order of derivatives are derived for this system. → Stability of Taylor's series method is studied. → Taylor's series method is A-stable for negative reactivity. → Taylor's series method is an accurate computational technique. - Abstract: Taylor's series method for solving the point reactor kinetics equations with multi-group of delayed neutrons in the presence of Newtonian temperature feedback reactivity is applied and programmed by FORTRAN. This system is the couples of the stiff nonlinear ordinary differential equations. This numerical method is based on the different order derivatives of the neutron density, the precursor concentrations of i-group of delayed neutrons and the reactivity. The r th order of derivatives are derived. The stability of Taylor's series method is discussed. Three sets of applications: step, ramp and temperature feedback reactivities are computed. Taylor's series method is an accurate computational technique and stable for negative step, negative ramp and temperature feedback reactivities. This method is useful than the traditional methods for solving the nonlinear point kinetics equations.

Integrodifferential equations and delay models in population dynamics

CERN Document Server

Cushing, Jim M

1977-01-01

These notes are, for the most part, the result of a course I taught at the University of Arizona during the Spring of 1977. Their main purpose is to inves­ tigate the effect that delays (of Volterra integral type) have when placed in the differential models of mathematical ecology, as far as stability of equilibria and the nature of oscillations of species densities are concerned. A secondary pur­ pose of the course out of which they evolved was to give students an (at least elementary) introduction to some mathematical modeling in ecology as well as to some purely mathematical subjects, such as stability theory for integrodifferentia1 systems, bifurcation theory, and some simple topics in perturbation theory. The choice of topics of course reflects my personal interests; and while these notes were not meant to exhaust the topics covered, I think they and the list of refer­ ences come close to covering the literature to date, as far as integrodifferentia1 models in ecology are concerned. I would like to th...

Moments of the Wigner delay times

International Nuclear Information System (INIS)

Berkolaiko, Gregory; Kuipers, Jack

2010-01-01

The Wigner time delay is a measure of the time spent by a particle inside the scattering region of an open system. For chaotic systems, the statistics of the individual delay times (whose average is the Wigner time delay) are thought to be well described by random matrix theory. Here we present a semiclassical derivation showing the validity of random matrix results. In order to simplify the semiclassical treatment, we express the moments of the delay times in terms of correlation functions of scattering matrices at different energies. In the semiclassical approximation, the elements of the scattering matrix are given in terms of the classical scattering trajectories, requiring one to study correlations between sets of such trajectories. We describe the structure of correlated sets of trajectories and formulate the rules for their evaluation to the leading order in inverse channel number. This allows us to derive a polynomial equation satisfied by the generating function of the moments. Along with showing the agreement of our semiclassical results with the moments predicted by random matrix theory, we infer that the scattering matrix is unitary to all orders in the semiclassical approximation.

Analysis of Caputo Impulsive Fractional Order Differential Equations with Applications

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Lakshman Mahto

2013-01-01

Full Text Available We use Sadovskii's fixed point method to investigate the existence and uniqueness of solutions of Caputo impulsive fractional differential equations of order with one example of impulsive logistic model and few other examples as well. We also discuss Caputo impulsive fractional differential equations with finite delay. The results proven are new and compliment the existing one.

A mathematical framework for functional mapping of complex phenotypes using delay differential equations.

Science.gov (United States)

Fu, Guifang; Wang, Zhong; Li, Jiahan; Wu, Rongling

2011-11-21

All biological phenomena occurring at different levels of organization from cells to organisms can be modeled as a dynamic system, in which the underlying components interact dynamically to comprehend its biological function. Such a systems modeling approach facilitates the use of biochemically and biophysically detailed mathematical models to describe and quantify "living cells," leading to an in-depth and precise understanding of the behavior, development and function of a biological system. Here, we illustrate how this approach can be used to map genes or quantitative trait loci (QTLs) that control a complex trait using the example of the circadian rhythm system which has been at the forefront of analytical mathematical modeling for many years. We integrate a system of biologically meaningful delay differential equations (DDEs) into functional mapping, a statistical model designed to map dynamic QTLs involved in biological processes. The DDEs model the ability of circadian rhythm to generate autonomously sustained oscillations with a period close to 24h, in terms of time-varying mRNA and protein abundances. By incorporating the Runge-Kutta fourth order algorithm within the likelihood-based context of functional mapping, we estimated the genetic parameters that define the periodic pattern of QTL effects on time-varying mRNA and protein abundances and their dynamic association as well as the linkage disequilibrium of the QTL and a marker. We prove theorems about how to choose appropriate parameters to guarantee periodic oscillations. We further used simulation studies to investigate how a QTL influences the period and the amplitude of circadian oscillations through changing model parameters. The model provides a quantitative framework for assessing the interplay between genetic effects of QTLs and rhythmic responses. Copyright © 2011 Elsevier Ltd. All rights reserved.

Knock Prediction Using a Simple Model for Ignition Delay

KAUST Repository

Kalghatgi, Gautam; Morganti, Kai; Algunaibet, Ibrahim; Sarathy, Mani; Dibble, Robert W.

2016-01-01

An earlier paper has shown the ability to predict the phasing of knock onset in a gasoline PFI engine using a simple ignition delay equation for an appropriate surrogate fuel made up of toluene and PRF (TPRF). The applicability of this approach

Stability result of the Timoshenko system with delay and boundary feedback

KAUST Repository

Said-Houari, Belkacem; Soufyane, Abdelaziz

2012-01-01

Our interest in this paper is to analyse the asymptotic behaviour of a Timoshenko beam system together with two boundary controls, with delay terms in the first and second equation. Assuming the weights of the delay are small enough, we show that the system is well-posed using the semigroup theory. Furthermore, we introduce a Lyapunov functional that gives the exponential decay of the total energy. © 2012 The author.

Stability result of the Timoshenko system with delay and boundary feedback

KAUST Repository

Said-Houari, Belkacem

2012-01-06

Our interest in this paper is to analyse the asymptotic behaviour of a Timoshenko beam system together with two boundary controls, with delay terms in the first and second equation. Assuming the weights of the delay are small enough, we show that the system is well-posed using the semigroup theory. Furthermore, we introduce a Lyapunov functional that gives the exponential decay of the total energy. © 2012 The author.

Crenelated fast oscillatory outputs of a two-delay electro-optic oscillator.

Science.gov (United States)

Weicker, Lionel; Erneux, Thomas; Jacquot, Maxime; Chembo, Yanne; Larger, Laurent

2012-02-01

An electro-optic oscillator subject to two distinct delayed feedbacks has been designed to develop pronounced broadband chaotic output. Its route to chaos starts with regular pulsating gigahertz oscillations that we investigate both experimentally and theoretically. Of particular physical interest are the transitions to various crenelated fast time-periodic oscillations, prior to the onset of chaotic regimes. The two-delay problem is described mathematically by two coupled delay-differential equations, which we analyze by using multiple-time-scale methods. We show that the interplay of a large delay and a relatively small delay is responsible for the onset of fast oscillations modulated by a slowly varying square-wave envelope. As the bifurcation parameter progressively increases, this envelope undergoes a sequence of bifurcations that corresponds to successive fixed points of a sine map.

A Heterogeneous Agent Model of Asspet Price with Three Time Delays

Directory of Open Access Journals (Sweden)

Akio Matsumoto

2016-09-01

Full Text Available This paper considers a continuous-time heterogeneous agent model ofa ...nancial market with one risky asset, two types of agents (i.e., thefundamentalists and the chartists, and three time delays. The chartistdemand is determined through a nonlinear function of the di¤erence be-tween the current price and a weighted moving average of the delayedprices whereas the fundamentalist demand is governed by the di¤erencebetween the current price and the fundamental value. The asset price dy-namics is described by a nonlinear delay di¤erential equation. Two mainresults are analytically and numerically shown:(i the delay destabilizes the market price and generates cyclic oscillationsaround the equilibrium;(ii under multiple delays, stability loss and gain repeatedly occurs as alength of the delay increases.

Finite-time stability of discrete fractional delay systems: Gronwall inequality and stability criterion

Science.gov (United States)

Wu, Guo-Cheng; Baleanu, Dumitru; Zeng, Sheng-Da

2018-04-01

This study investigates finite-time stability of Caputo delta fractional difference equations. A generalized Gronwall inequality is given on a finite time domain. A finite-time stability criterion is proposed for fractional differential equations. Then the idea is extended to the discrete fractional case. A linear fractional difference equation with constant delays is considered and finite-time stable conditions are provided. One example is numerically illustrated to support the theoretical result.

Delay differential systems for tick population dynamics.

Science.gov (United States)

Fan, Guihong; Thieme, Horst R; Zhu, Huaiping

2015-11-01

Ticks play a critical role as vectors in the transmission and spread of Lyme disease, an emerging infectious disease which can cause severe illness in humans or animals. To understand the transmission dynamics of Lyme disease and other tick-borne diseases, it is necessary to investigate the population dynamics of ticks. Here, we formulate a system of delay differential equations which models the stage structure of the tick population. Temperature can alter the length of time delays in each developmental stage, and so the time delays can vary geographically (and seasonally which we do not consider). We define the basic reproduction number [Formula: see text] of stage structured tick populations. The tick population is uniformly persistent if [Formula: see text] and dies out if [Formula: see text]. We present sufficient conditions under which the unique positive equilibrium point is globally asymptotically stable. In general, the positive equilibrium can be unstable and the system show oscillatory behavior. These oscillations are primarily due to negative feedback within the tick system, but can be enhanced by the time delays of the different developmental stages.

Incorporating time-delays in S-System model for reverse engineering genetic networks.

Science.gov (United States)

Chowdhury, Ahsan Raja; Chetty, Madhu; Vinh, Nguyen Xuan

2013-06-18

In any gene regulatory network (GRN), the complex interactions occurring amongst transcription factors and target genes can be either instantaneous or time-delayed. However, many existing modeling approaches currently applied for inferring GRNs are unable to represent both these interactions simultaneously. As a result, all these approaches cannot detect important interactions of the other type. S-System model, a differential equation based approach which has been increasingly applied for modeling GRNs, also suffers from this limitation. In fact, all S-System based existing modeling approaches have been designed to capture only instantaneous interactions, and are unable to infer time-delayed interactions. In this paper, we propose a novel Time-Delayed S-System (TDSS) model which uses a set of delay differential equations to represent the system dynamics. The ability to incorporate time-delay parameters in the proposed S-System model enables simultaneous modeling of both instantaneous and time-delayed interactions. Furthermore, the delay parameters are not limited to just positive integer values (corresponding to time stamps in the data), but can also take fractional values. Moreover, we also propose a new criterion for model evaluation exploiting the sparse and scale-free nature of GRNs to effectively narrow down the search space, which not only reduces the computation time significantly but also improves model accuracy. The evaluation criterion systematically adapts the max-min in-degrees and also systematically balances the effect of network accuracy and complexity during optimization. The four well-known performance measures applied to the experimental studies on synthetic networks with various time-delayed regulations clearly demonstrate that the proposed method can capture both instantaneous and delayed interactions correctly with high precision. The experiments carried out on two well-known real-life networks, namely IRMA and SOS DNA repair network in

TWIN POSITIVE PERIODIC SOLUTIONS OF FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

Institute of Scientific and Technical Information of China (English)

无

2006-01-01

In this paper, the author studies a class of nonlinear functional differential equation. By using a fixed point theorem in cones, sufficient conditions are established for the existence of twin positive periodic solutions.

Steady State Analysis of Stochastic Systems with Multiple Time Delays

Science.gov (United States)

Xu, W.; Sun, C. Y.; Zhang, H. Q.

In this paper, attention is focused on the steady state analysis of a class of nonlinear dynamic systems with multi-delayed feedbacks driven by multiplicative correlated Gaussian white noises. The Fokker-Planck equations for delayed variables are at first derived by Novikov's theorem. Then, under small delay assumption, the approximate stationary solutions are obtained by the probability density approach. As a special case, the effects of multidelay feedbacks and the correlated additive and multiplicative Gaussian white noises on the response of a bistable system are considered. It is shown that the obtained analytical results are in good agreement with experimental results in Monte Carlo simulations.

Stability analysis of solutions to nonlinear stiff Volterra functional differential equations in Banach spaces

Institute of Scientific and Technical Information of China (English)

LI Shoufu

2005-01-01

A series of stability, contractivity and asymptotic stability results of the solutions to nonlinear stiff Volterra functional differential equations (VFDEs) in Banach spaces is obtained, which provides the unified theoretical foundation for the stability analysis of solutions to nonlinear stiff problems in ordinary differential equations(ODEs), delay differential equations(DDEs), integro-differential equations(IDEs) and VFDEs of other type which appear in practice.

Comment on ‘Time delays in molecular photoionization’

Science.gov (United States)

Baykusheva, Denitsa; Wörner, Hans Jakob

2017-04-01

In a recent article by Hockett et al (2016 J. Phys. B: At. Mol. Opt. Phys. 49 095602), time delays arising in the context of molecular single-photon ionization are investigated from a theoretical point of view. We argue that one of the central equations given in this article is incorrect and present a reformulation that is consistent with the established treatment of angle-dependent scattering delays (Eisenbud 1948 PhD Thesis Princeton University; Wigner 1955 Phys. Rev. 98 145-7 Smith 1960 Phys. Rev. 118 349-6 Nussenzveig 1972 Phys. Rev. D 6 1534-42).

Delay-induced stochastic bifurcations in a bistable system under white noise

International Nuclear Information System (INIS)

Sun, Zhongkui; Fu, Jin; Xu, Wei; Xiao, Yuzhu

2015-01-01

In this paper, the effects of noise and time delay on stochastic bifurcations are investigated theoretically and numerically in a time-delayed Duffing-Van der Pol oscillator subjected to white noise. Due to the time delay, the random response is not Markovian. Thereby, approximate methods have been adopted to obtain the Fokker-Planck-Kolmogorov equation and the stationary probability density function for amplitude of the response. Based on the knowledge that stochastic bifurcation is characterized by the qualitative properties of the steady-state probability distribution, it is found that time delay and feedback intensity as well as noise intensity will induce the appearance of stochastic P-bifurcation. Besides, results demonstrated that the effects of the strength of the delayed displacement feedback on stochastic bifurcation are accompanied by the sensitive dependence on time delay. Furthermore, the results from numerical simulations best confirm the effectiveness of the theoretical analyses

Delay-induced stochastic bifurcations in a bistable system under white noise

Energy Technology Data Exchange (ETDEWEB)

Sun, Zhongkui, E-mail: sunzk@nwpu.edu.cn; Fu, Jin; Xu, Wei [Department of Applied Mathematics, Northwestern Polytechnical University, Xi' an 710072 (China); Xiao, Yuzhu [Department of Mathematics and Information Science, Chang' an University, Xi' an 710086 (China)

2015-08-15

In this paper, the effects of noise and time delay on stochastic bifurcations are investigated theoretically and numerically in a time-delayed Duffing-Van der Pol oscillator subjected to white noise. Due to the time delay, the random response is not Markovian. Thereby, approximate methods have been adopted to obtain the Fokker-Planck-Kolmogorov equation and the stationary probability density function for amplitude of the response. Based on the knowledge that stochastic bifurcation is characterized by the qualitative properties of the steady-state probability distribution, it is found that time delay and feedback intensity as well as noise intensity will induce the appearance of stochastic P-bifurcation. Besides, results demonstrated that the effects of the strength of the delayed displacement feedback on stochastic bifurcation are accompanied by the sensitive dependence on time delay. Furthermore, the results from numerical simulations best confirm the effectiveness of the theoretical analyses.

A necessary and sufficient condition for well-posedness of initial value problems of retarded functional differential equations

Science.gov (United States)

Nishiguchi, Junya

2017-09-01

We introduce the retarded functional differential equations (RFDEs) with general delay structure to treat various delay differential equations (DDEs) in a unified way and to clarify the delay structure in those dynamics. We are interested in the question as to which space of histories is suitable for the dynamics of each DDE, and investigate the well-posedness of the initial value problems (IVPs) of the RFDEs. A main theorem is that the IVP is well-posed for any ;admissible; history functional if and only if the semigroup determined by the trivial RFDE x ˙ = 0 is continuous. We clarify the meaning of the Hale-Kato axiom (Hale & Kato [12]) by applying this result to RFDEs with infinite delay. We also apply the result to DDEs with unbounded time- and state-dependent delays.

Empirical Sampling Distributions of Equating Coefficients for Graded and Nominal Response Instruments.

Science.gov (United States)

Baker, Frank B.

1997-01-01

Examined the sampling distributions of equating coefficients produced by the characteristic curve method for tests using graded and nominal response scoring using simulated data. For both models and across all three equating situations, the sampling distributions were generally bell-shaped and peaked, and occasionally had a small degree of…

Determinants of patient delay in seeking treatment among ...

African Journals Online (AJOL)

lmboera

A descriptive cross sectional study was employed. A face to .... Face validation of the instrument was done through peer review and pretesting. In this ..... patient delay for Christians was about 3.3 fold higher compared with Muslims. ... Grover, A., Kumar, R. & Jindal, K. (2006) Socio-demographic determinants of treatment.

Time delay and noise explaining the behaviour of the cell growth in fermentation process

Energy Technology Data Exchange (ETDEWEB)

Ayuobi, Tawfiqullah; Rosli, Norhayati [Faculty of Industrial Sciences and Technology, Universiti Malaysia Pahang, Lebuhraya Tun Razak, 26300 Gambang, Pahang (Malaysia); Bahar, Arifah [Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor (Malaysia); Salleh, Madihah Md [Department of Biotechnology Industry, Faculty of Biosciences and Bioengineering, Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor (Malaysia)

2015-02-03

This paper proposes to investigate the interplay between time delay and external noise in explaining the behaviour of the microbial growth in batch fermentation process. Time delay and noise are modelled jointly via stochastic delay differential equations (SDDEs). The typical behaviour of cell concentration in batch fermentation process under this model is investigated. Milstein scheme is applied for solving this model numerically. Simulation results illustrate the effects of time delay and external noise in explaining the lag and stationary phases, respectively for the cell growth of fermentation process.

Time delay and noise explaining the behaviour of the cell growth in fermentation process

Science.gov (United States)

Ayuobi, Tawfiqullah; Rosli, Norhayati; Bahar, Arifah; Salleh, Madihah Md

2015-02-01

This paper proposes to investigate the interplay between time delay and external noise in explaining the behaviour of the microbial growth in batch fermentation process. Time delay and noise are modelled jointly via stochastic delay differential equations (SDDEs). The typical behaviour of cell concentration in batch fermentation process under this model is investigated. Milstein scheme is applied for solving this model numerically. Simulation results illustrate the effects of time delay and external noise in explaining the lag and stationary phases, respectively for the cell growth of fermentation process.

Time delay and noise explaining the behaviour of the cell growth in fermentation process

International Nuclear Information System (INIS)

Ayuobi, Tawfiqullah; Rosli, Norhayati; Bahar, Arifah; Salleh, Madihah Md

2015-01-01

This paper proposes to investigate the interplay between time delay and external noise in explaining the behaviour of the microbial growth in batch fermentation process. Time delay and noise are modelled jointly via stochastic delay differential equations (SDDEs). The typical behaviour of cell concentration in batch fermentation process under this model is investigated. Milstein scheme is applied for solving this model numerically. Simulation results illustrate the effects of time delay and external noise in explaining the lag and stationary phases, respectively for the cell growth of fermentation process

D-leaping: Accelerating stochastic simulation algorithms for reactions with delays

International Nuclear Information System (INIS)

Bayati, Basil; Chatelain, Philippe; Koumoutsakos, Petros

2009-01-01

We propose a novel, accelerated algorithm for the approximate stochastic simulation of biochemical systems with delays. The present work extends existing accelerated algorithms by distributing, in a time adaptive fashion, the delayed reactions so as to minimize the computational effort while preserving their accuracy. The accuracy of the present algorithm is assessed by comparing its results to those of the corresponding delay differential equations for a representative biochemical system. In addition, the fluctuations produced from the present algorithm are comparable to those from an exact stochastic simulation with delays. The algorithm is used to simulate biochemical systems that model oscillatory gene expression. The results indicate that the present algorithm is competitive with existing works for several benchmark problems while it is orders of magnitude faster for certain systems of biochemical reactions.

Hopf bifurcation for tumor-immune competition systems with delay

Directory of Open Access Journals (Sweden)

Ping Bi

2014-01-01

Full Text Available In this article, a immune response system with delay is considered, which consists of two-dimensional nonlinear differential equations. The main purpose of this paper is to explore the Hopf bifurcation of a immune response system with delay. The general formula of the direction, the estimation formula of period and stability of bifurcated periodic solution are also given. Especially, the conditions of the global existence of periodic solutions bifurcating from Hopf bifurcations are given. Numerical simulations are carried out to illustrate the the theoretical analysis and the obtained results.

Stability switches, oscillatory multistability, and spatio-temporal patterns of nonlinear oscillations in recurrently delay coupled neural networks.

Science.gov (United States)

Song, Yongli; Makarov, Valeri A; Velarde, Manuel G

2009-08-01

A model of time-delay recurrently coupled spatially segregated neural assemblies is here proposed. We show that it operates like some of the hierarchical architectures of the brain. Each assembly is a neural network with no delay in the local couplings between the units. The delay appears in the long range feedforward and feedback inter-assemblies communications. Bifurcation analysis of a simple four-units system in the autonomous case shows the richness of the dynamical behaviors in a biophysically plausible parameter region. We find oscillatory multistability, hysteresis, and stability switches of the rest state provoked by the time delay. Then we investigate the spatio-temporal patterns of bifurcating periodic solutions by using the symmetric local Hopf bifurcation theory of delay differential equations and derive the equation describing the flow on the center manifold that enables us determining the direction of Hopf bifurcations and stability of the bifurcating periodic orbits. We also discuss computational properties of the system due to the delay when an external drive of the network mimicks external sensory input.

Workshop on Numerical Methods for Ordinary Differential Equations

CERN Document Server

Gear, Charles; Russo, Elvira

1989-01-01

Developments in numerical initial value ode methods were the focal topic of the meeting at L'Aquila which explord the connections between the classical background and new research areas such as differental-algebraic equations, delay integral and integro-differential equations, stability properties, continuous extensions (interpolants for Runge-Kutta methods and their applications, effective stepsize control, parallel algorithms for small- and large-scale parallel architectures). The resulting proceedings address many of these topics in both research and survey papers.

New exponential stability conditions for linear delayed systems of differential equations

Directory of Open Access Journals (Sweden)

Leonid Berezansky

2016-08-01

where $t\\ge 0$, $m$ and $r_{ij}$, $i,j=1,\\dots,m$ are natural numbers, $a_{ij}^{k}\\colon [0,\\infty\\to\\mathbb{R}$ are measurable coefficients, and $h_{ij}^{k}\\colon [0,\\infty\\to\\mathbb{R}$ are measurable delays. The progress was achieved by using a new technique making it possible to replace the constant $1$ by the constant $1+{1}/{\\mathrm{e}}$ on the right-hand sides of crucial inequalities ensuring exponential stability.

An efficient technique for the point reactor kinetics equations with Newtonian temperature feedback effects

International Nuclear Information System (INIS)

Nahla, Abdallah A.

2011-01-01

Highlights: → An efficient technique for the nonlinear reactor kinetics equations is presented. → This method is based on Backward Euler or Crank Nicholson and fundamental matrix. → Stability of efficient technique is defined and discussed. → This method is applied to point kinetics equations of six-groups of delayed neutrons. → Step, ramp, sinusoidal and temperature feedback reactivities are discussed. - Abstract: The point reactor kinetics equations of multi-group of delayed neutrons in the presence Newtonian temperature feedback effects are a system of stiff nonlinear ordinary differential equations which have not any exact analytical solution. The efficient technique for this nonlinear system is based on changing this nonlinear system to a linear system by the predicted value of reactivity and solving this linear system using the fundamental matrix of the homogenous linear differential equations. The nonlinear point reactor kinetics equations are rewritten in the matrix form. The solution of this matrix form is introduced. This solution contains the exponential function of a variable coefficient matrix. This coefficient matrix contains the unknown variable, reactivity. The predicted values of reactivity in the explicit form are determined replacing the exponential function of the coefficient matrix by two kinds, Backward Euler and Crank Nicholson, of the rational approximations. The nonlinear point kinetics equations changed to a linear system of the homogenous differential equations. The fundamental matrix of this linear system is calculated using the eigenvalues and the corresponding eigenvectors of the coefficient matrix. Stability of the efficient technique is defined and discussed. The efficient technique is applied to the point kinetics equations of six-groups of delayed neutrons with step, ramp, sinusoidal and the temperature feedback reactivities. The results of these efficient techniques are compared with the traditional methods.

Delay time in a single barrier for a movable quantum shutter

International Nuclear Information System (INIS)

Hernandez, Alberto

2010-01-01

The transient solution and delay time for a δ potential scatterer with a movable quantum shutter is calculated by solving analytically the time-dependent Schroedinger equation. The delay time is analyzed as a function of the distance between the shutter and the potential barrier and also as a function of the distance between the potential barrier and the detector. In both cases, it is found that the delay time exhibits a dynamical behavior and that it tends to a saturation value Δt sat in the limit of very short distances, which represents the maximum delay produced by the potential barrier near the interaction region. The phase time τ θ , on the other hand, is not an appropriate time scale for measuring the time delay near the interaction region, except if the shutter is moved far away from the potential. The role played by the antibound state of the system on the behavior of the delay time is also discussed.

Asymptotic behavior of second-order impulsive differential equations

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Haifeng Liu

2011-02-01

Full Text Available In this article, we study the asymptotic behavior of all solutions of 2-th order nonlinear delay differential equation with impulses. Our main tools are impulsive differential inequalities and the Riccati transformation. We illustrate the results by an example.

Nonlinear Estimation of Discrete-Time Signals Under Random Observation Delay

International Nuclear Information System (INIS)

Caballero-Aguila, R.; Jimenez-Lopez, J. D.; Hermoso-Carazo, A.; Linares-Perez, J.; Nakamori, S.

2008-01-01

This paper presents an approximation to the nonlinear least-squares estimation problem of discrete-time stochastic signals using nonlinear observations with additive white noise which can be randomly delayed by one sampling time. The observation delay is modelled by a sequence of independent Bernoulli random variables whose values, zero or one, indicate that the real observation arrives on time or it is delayed and, hence, the available measurement to estimate the signal is not up-to-date. Assuming that the state-space model generating the signal is unknown and only the covariance functions of the processes involved in the observation equation are ready for use, a filtering algorithm based on linear approximations of the real observations is proposed.

On global stability criterion for neural networks with discrete and distributed delays

International Nuclear Information System (INIS)

Park, Ju H.

2006-01-01

Based on the Lyapunov functional stability analysis for differential equations and the linear matrix inequality (LMI) optimization approach, a new delay-dependent criterion for neural networks with discrete and distributed delays is derived to guarantee global asymptotic stability. The criterion is expressed in terms of LMIs, which can be solved easily by various convex optimization algorithms. Some numerical examples are given to show the effectiveness of proposed method

Stability and bifurcation of a discrete BAM neural network model with delays

International Nuclear Information System (INIS)

Zheng Baodong; Zhang Yang; Zhang Chunrui

2008-01-01

A map modelling a discrete bidirectional associative memory neural network with delays is investigated. Its dynamics is studied in terms of local analysis and Hopf bifurcation analysis. By analyzing the associated characteristic equation, its linear stability is investigated and Hopf bifurcations are demonstrated. It is found that there exist Hopf bifurcations when the delay passes a sequence of critical values. Numerical simulation is performed to verify the analytical results

A note on exponential convergence of neural networks with unbounded distributed delays

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Chu Tianguang [Intelligent Control Laboratory, Center for Systems and Control, Department of Mechanics and Engineering Science, Peking University, Beijing 100871 (China)]. E-mail: chutg@pku.edu.cn; Yang Haifeng [Intelligent Control Laboratory, Center for Systems and Control, Department of Mechanics and Engineering Science, Peking University, Beijing 100871 (China)

2007-12-15

This note examines issues concerning global exponential convergence of neural networks with unbounded distributed delays. Sufficient conditions are derived by exploiting exponentially fading memory property of delay kernel functions. The method is based on comparison principle of delay differential equations and does not need the construction of any Lyapunov functionals. It is simple yet effective in deriving less conservative exponential convergence conditions and more detailed componentwise decay estimates. The results of this note and [Chu T. An exponential convergence estimate for analog neural networks with delay. Phys Lett A 2001;283:113-8] suggest a class of neural networks whose globally exponentially convergent dynamics is completely insensitive to a wide range of time delays from arbitrary bounded discrete type to certain unbounded distributed type. This is of practical interest in designing fast and reliable neural circuits. Finally, an open question is raised on the nature of delay kernels for attaining exponential convergence in an unbounded distributed delayed neural network.