On Step-Like Contrast Structure of Singularly Perturbed Systems

Mingkang Ni,Zhiming Wang

doi:10.1155/2009/634324

Abstract

The existence of a step-like contrast structure for a class of high-dimensional singularly perturbed system is shown by a smooth connection method based on the existence of a first integral for an associated system. In the framework of this paper, we not only give the conditions under which there exists an internal transition layer but also determine where an internal transition time is. Meanwhile, the uniformly valid asymptotic expansion of a solution with a step-like contrast structure is presented.

Highlights

The problem of contrast structures is a singularly perturbed problem whose solutions with both internal transition layers and boundary layers
Most works on internal layer solutions concentrate on singularly perturbed parabolic systems by geometric method see 1 and the references therein
In Russia, the works on singularly perturbed ordinary equations are concerned by boundary function method 2–5

Summary

Introduction

The problem of contrast structures is a singularly perturbed problem whose solutions with both internal transition layers and boundary layers. For a high dimensional singularly perturbed system, we cannot always find such an equation like 1.9 to determine t0 at which there exists a heteroclinic orbit This is one difficulty to further study the problem on step-like contrast structures. The existence of a homoclinic or heteroclinic orbit in high dimension space and how to construct such an orbit are themselves open in general in the qualitative analysis geometric method theory 8–10 To explore these high dimensional contrast structure problems, we just start from some particular class of singularly perturbed system and are trying to develop some approach to construct a desired heteroclinic orbit by using a first integral method for such a class of the system and determine its internal transition time t0

Problem Formulation

Construction of Asymptotic Solution

Existence of Step-Like Solution and Its Limit Theorem

Conclusive Remarks