Using Melnikov's method to solve Silnikov's problems

1990 ◽  
Vol 116 (3-4) ◽  
pp. 295-325 ◽  
Author(s):  
Xiao-Biao Lin
Keyword(s):  
Differential Equation ◽  
Function Space ◽  
Delay Differential Equation ◽  
Homoclinic Solutions ◽  
Singularly Perturbed ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Delay Differential ◽  
Limiting Case ◽  
Space Approach

SynopsisA function space approach is employed to obtain bifurcation functions for which the zeros correspond to the occurrence of periodic or aperiodic solutions near heteroclinic or homoclinic cycles. The bifurcation function for the existence of homoclinic solutions is the limiting case where the period is infinite. Examples include generalisations of Silnikov's main theorems and a retreatment of a singularly perturbed delay differential equation.

Periodic solutions of a singularly perturbed delay differential equation

2008 ◽  
Vol 237 (24) ◽  
pp. 3307-3321 ◽  
Author(s):  
Mohit H. Adhikari ◽  
Evangelos A. Coutsias ◽  
John K. McIver
Keyword(s):  
Differential Equation ◽  
Periodic Solutions ◽  
Delay Differential Equation ◽  
Singularly Perturbed ◽  
Delay Differential

scholarly journals Solution Of Singularly Perturbed Delay Differential Equations Using Liouville Green Transformation

2021 ◽  
Vol 12 (4) ◽  
pp. 325-335
Author(s):  
M. Adilaxmi , Et. al.
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Perturbation Problem ◽  
Delay Differential ◽  
Singularly Perturbation ◽  
The Given

This paper envisages the use of Liouville Green Transformation to find the solution of singularly perturbed delay differential equations. First, using Taylor series, the given singularly perturbed delay differential equation is approximated by an asymptotically equivalent singularly perturbation problem. Then the Liouville Green Transformation is applied to get the solution. The method is demonstrated by implementing several model examples by taking various values for the delay parameter and perturbation parameter.

scholarly journals Quintic B-Spline Technique for Numerical Treatment of Third Order Singular Perturbed Delay Differential Equation

2019 ◽  
Vol 4 (6) ◽  
pp. 1471-1482
Author(s):  
Mandeep Kaur Vaid ◽  
Geeta Arora
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Numerical Treatment ◽  
Spline Collocation ◽  
Third Order ◽  
B Spline ◽  
Collocation Technique ◽  
Delay Differential

In this paper, a class of third order singularly perturbed delay differential equation with large delay is considered for numerical treatment. The considered equation has discontinuous convection-diffusion coefficient and source term. A quintic trigonometric B-spline collocation technique is used for numerical simulation of the considered singularly perturbed delay differential equation by dividing the domain into the uniform mesh. Further, uniform convergence of the solution is discussed by using the concept of Hall error estimation and the method is found to be of first-order convergent. The existence of the solution is also established. Computation work is carried out to validate the theoretical results.

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