scholarly journals Dependence of Dynamics of a System of Two Coupled Generators with Delayed Feedback on the Sign of Coupling

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1790
Author(s):  
Alexandra Kashchenko
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Coupling Parameter ◽  
Delayed Feedback ◽  
Structure Of Solutions ◽  
Asymptotics Of Solutions ◽  
Finite Dimensional ◽  
Method Of Steps ◽  
Delay Differential ◽  
Negative Coupling

In this paper, we study the nonlocal dynamics of a system of delay differential equations with large parameters. This system simulates coupled generators with delayed feedback. Using the method of steps, we construct asymptotics of solutions. By these asymptotics, we construct a special finite-dimensional map. This map helps us to determine the structure of solutions. We study the dependence of solutions on the coupling parameter and show that the dynamics of the system is significantly different in the case of positive coupling and in the case of negative coupling.

scholarly journals Constructive solution of coupled second order delay differential equations with variable coefficients

1995 ◽  
Vol 18 (4) ◽  
pp. 689-700
Author(s):  
R. J. Villanueva ◽  
A. Hervas ◽  
M. V. Ferrer
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Bounded Domain ◽  
Delay Differential Equations ◽  
Approximation Error ◽  
Variable Coefficients ◽  
Second Order ◽  
Initial Value ◽  
Method Of Steps ◽  
Delay Differential

In this paper, we study initial value problems for coupld second order delay differential equations with variable coefficients. By means of the application of the method of steps and the method of Frobenius, the exact solution of the problem is constrcted. Then, in a bounded domain, a finite analytic solution with error bounds is provided. Given an admissible errorϵwe give the number of terms to be taken in the infinite series exact solution so that the approximation error be smaller than in the bounded domain.

Galerkin Projections for Delay Differential Equations

2004 ◽  
Vol 127 (1) ◽  
pp. 80-87 ◽  
Author(s):  
Pankaj Wahi ◽  
Anindya Chatterjee
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Galerkin Projection ◽  
Shape Functions ◽  
Projection Technique ◽  
Finite Dimensional ◽  
Systems Of Odes ◽  
Qualitative Dynamics ◽  
Delay Differential ◽  
Galerkin Projections

We present a Galerkin projection technique by which finite-dimensional ordinary differential equation (ODE) approximations for delay differential equations (DDEs) can be obtained in a straightforward fashion. The technique requires neither the system to be near a bifurcation point, nor the delayed terms to have any specific restrictive form, or even the delay, nonlinearities, and/or forcing to be small. We show through several numerical examples that the systems of ODEs obtained using this procedure can accurately capture the dynamics of the DDEs under study, and that the accuracy of solutions increases with increasing numbers of shape functions used in the Galerkin projection. Examples studied here include a linear constant coefficient DDE as well as forced nonlinear DDEs with one or more delays and possibly nonlinear delayed terms. Parameter studies, with associated bifurcation diagrams, show that the qualitative dynamics of the DDEs can be captured satisfactorily with a modest number of shape functions in the Galerkin projection.

A New Look at the Stability Analysis of Delay-Differential Equations

2005 ◽  
Author(s):  
Tama´s Kalma´r-Nagy
Keyword(s):  
Difference Equation ◽  
Differential Equations ◽  
Characteristic Equation ◽  
Delay Differential Equations ◽  
Linear Delay ◽  
The Laplace Transform ◽  
Method Of Steps ◽  
The Difference ◽  
The Stability ◽  
Delay Differential

It is shown that the method of steps for linear delay-differential equations combined with the Laplace-transform can be used to determine the stability of the equation. The result of the method is an infinite dimensional difference equation whose stability corresponds to that of the transcendental characteristic equation. Truncations of this difference equation are used to construct numerical stability charts. The method is demonstrated on a first and second order delay equation. Correspondence between the transcendental characteristic equation and the difference equation is proved for the first order case.

scholarly journals Computation of double Hopf points for delay differential equations

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Yingxiang Xu ◽  
Tingting Shi
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Delay Systems ◽  
System Of Equations ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Defining System ◽  
Numerical Bifurcation ◽  
Delay Differential ◽  
Branch Switching

AbstractRelating to the crucial problem of branch switching, the calculation of codimension 2 bifurcation points is one of the major issues in numerical bifurcation analysis. In this paper, we focus on the double Hopf points for delay differential equations and analyze in detail the corresponding eigenspace, which enable us to obtain the finite dimensional defining system of equations of such points, instead of an infinite dimensional one that happens naturally for delay systems. We show that the double Hopf point, together with the corresponding eigenvalues, eigenvectors and the critical values of the bifurcation parameters, is a regular solution of the finite dimensional defining system of equations, and thus can be obtained numerically through applying the classical iterative methods. We show our theoretical findings by a numerical example.

scholarly journals Hermite Wavelet Method for Fractional Delay Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Umer Saeed ◽  
Mujeeb ur Rehman
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Wavelet Method ◽  
Numerical Examples ◽  
Method Of Steps ◽  
Fractional Delay ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

We proposed a method by utilizing method of steps and Hermite wavelet method, for solving the fractional delay differential equations. This technique first converts the fractional delay differential equation to a fractional nondelay differential equation and then applies the Hermite wavelet method on the obtained fractional nondelay differential equation to find the solution. Several numerical examples are solved to show the applicability of the proposed method.

ALMOST SURE ASYMPTOTIC STABILITY OF SCALAR STOCHASTIC DELAY EQUATIONS: FINITE STATE MARKOV PROCESS

2012 ◽  
Vol 12 (01) ◽  
pp. 1150010 ◽  
Author(s):  
N. SRI NAMACHCHIVAYA ◽  
VOLKER WIHSTUTZ
Keyword(s):  
Asymptotic Stability ◽  
Markov Process ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Almost Sure Stability ◽  
Stochastic Delay ◽  
Finite Dimensional ◽  
Finite State ◽  
Parametric Fluctuations ◽  
Delay Differential

In this paper, we study the almost-sure asymptotic stability of scalar delay differential equations with random parametric fluctuations which are modeled by a Markov process with finitely many states. The techniques developed for the determination of almost-sure asymptotic stability of finite dimensional stochastic differential equations will be extended to delay differential equations with random parametric fluctuations. For small intensity noise, we construct an asymptotic expansion for the exponential growth rate (the maximal Lyapunov exponent), which determines the almost-sure stability of the stochastic system.

Sign in / Sign up

Export Citation Format

Share Document