Some Results On Contrast Structures For Higher Dimensional Singularly Perturbed Systems

The thesis aims to investigate the contrast structure for higher dimen-sional singularly perturbed systems with multiple scales, including Tikhonov system and singular singularly perturbed system. In recent years, the study on the contrast structure becomes a hot topic in the singularly perturbed problems. When we deal with the singularly perturbed problems, we often encounter the contrast structure, such as the analysis about the deformation law at the moment of the automobiles colliding each other in vehicle collision modles, the transfer law of the neurons in neural network models and the boundary layer phenomena in superconductor surface. In fact, many phe-nomena in singularly perturbed problems are closely related to the contrast structure, including multi-layers, embedded-layers and non-exponential de-cay of boundary functions, etc. When the reduced system has several isolate roots and the solution of the original problem approaches different reduced root in different time, the contrast structure occurs. The structure of such solutions is very complex. As we know, the contrast structure in singularly perturbed problems is mainly classified as a step-type contrast structure or a spike-type contrast structure which corresponds to heteroclinic orbit or ho-moclinic orbit in their phase space. Its fundamental characteristic is that there is a t*(or multiple t*) within the domain of interest, which is called as an internal transition point. The discussion at each l*is exactly the same, so we only study the case that there exists one internal transition point. The position of t*is unknown in advance and it needs to be determined thereafter. In the neighborhood of t*, the solution y(t,u) will have an abrupt structure change. In the different sides of t*, if y(t,u) approaches different re-duced solutions, we call it step-type contrast structure. If y(t,u) approaches to the same reduced solution, we call it spike-type contrast structure. Our objective in this paper is to study the contrast structures for Tikhonov sys-tem and singular singularly perturbed system. By means of the first integral we find a higher dimensional heteroclinic orbit in need in a fast phase space. The formal asymptotic solution is constructed by boundary function method, and the internal transition time t*is determined when we solve the boundary functions. Using the method of sewing connection, differential inequality and k+σ exchange lemma we obtain the uniformly valid asymptotical expansion of such an available step-like contrast structure.This dissertation is divided into three parts. The first part considers the contrast structure for higher dimensional singularly perturbed systems. The second part considers the contrast structure for singular singularly perturbed system and the last part gives an application for the contrast structure in differential-difference equation.The main research results are outlined as follows:Chapter One introduces the history and actuality for the singular per-turbation and the contrast structure, gives some basic concepts and lemmas which are relevant to our study, and elaborates the main research results and innovative points in this paper.Chapter Two studies the contrast structure for the quasi-linear equation with slow variable. The asymptotic solution of this problem is constructed by boundary function method. By sewing orbit smooth, the existence of the step-like contrast structure is shown and the asymptotic solution is proved to be uniformly effective in the whole interval.Chapter Three is devoted to investigate the contrast structure for (M+m) dimensional Tikhonov system with initial-boundary value conditions. By means of the first integral we find a higher dimensional hetero clinic orbit in need in a fast phase space. The formal asymptotic solution is constructed by boundary function method. The internal transition time t*is determined when we solve the higher order boundary functions, where we use the proper-ties of exponential dichotomies and the Fredholm alternatives. The uniformly valid asymptotic expansion and the existence of such an available step-like contrast structure are obtained by sewing connection method.The research of Chapter Four is the contrast structure for (M+m) dimen-sional Tikhonov system with boundary value conditions. In this chapter, we combine the boundary function method with the theory of geometric singular perturbation. By means of the first integral we find a higher dimensional het-eroclinic orbit in need in a fast phase space. The internal transition time t*is determined when we solve the higher order boundary functions, where we use the properties of exponential dichotomies and the Fredholm alternatives. We construct the formal asymptotic solution by boundary function method, and using the method of k+σ changing lemma we prove the existence of the step-like contrast structure. At the same time, the asymptotic solution is proved to be uniformly effective in the whole interval.In Chapter Five, we focus on the contrast structure for higher dimen-sional singular singularly perturbed system. In this chapter, we combine the boundary function method with the theory of geometric singular perturba- tion. By means of the first integral we find a higher dimensional heteroclinic orbit in need in a fast phase space and t*is determined at the same time. Using the method of boundary function, we construct the formal asymptotic solution. The existence of the step-like contrast structure is proved by k+σ exchange lemma. At the same time, the asymptotic solution is proved to be uniformly effective in the whole interval.In Chapter Six, the interior layer for a second order nonlinear singularly perturbed differential-difference equation is shown. Using the method of boundary function and fractional steps, we construct the formal asymptotic expansion and point out that the boundary layer at t=0has a great influence upon the interior layer at t=σ. At the same time, based on differential inequality techniques, the existence of the solution and the uniform validity of the asymptotic expansion are proved.