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.

Galerkin Projections for Delay Differential Equations

2003 ◽  
Author(s):  
Pankaj Wahi ◽  
Anindya Chatterjee
Keyword(s):  
Delay Differential Equations ◽  
Galerkin Projection ◽  
Bifurcation Diagrams ◽  
Shape Functions ◽  
Projection Technique ◽  
Numerical Examples ◽  
Finite Dimensional ◽  
Qualitative Dynamics ◽  
Delay Differential ◽  
Galerkin Projections

We present a Galerkin projection technique by which finite-dimensional ODE approximations for DDE’s 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, nor even the delay, nonlinearities and/or forcing to be small. We show through several numerical examples that the systems of ODE’s obtained using this procedure can accurately capture the dynamics of the DDE’s 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 DDE’s with one or more delays and possibly nonlinear delayed terms. Parameter studies, with associated bifurcation diagrams, show that the qualitative dynamics of the DDE’s can be captured satisfactorily with a modest number of shape functions in the Galerkin projection.

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 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.

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.

A new Ф-generalized quasi metric space with some fixed point results and applications

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3157-3172
Author(s):  
Mujahid Abbas ◽  
Bahru Leyew ◽  
Safeer Khan
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Points ◽  
Periodic Solutions ◽  
Metric Space ◽  
Delay Differential Equations ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Existence Of Periodic Solutions ◽  
Delay Differential

In this paper, the concept of a new ?-generalized quasi metric space is introduced. A number of well-known quasi metric spaces are retrieved from ?-generalized quasi metric space. Some general fixed point theorems in a ?-generalized quasi metric spaces are proved, which generalize, modify and unify some existing fixed point theorems in the literature. We also give applications of our results to obtain fixed points for contraction mappings in the domain of words and to prove the existence of periodic solutions of delay differential equations.

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