摘要
本文旨在研究发生在具有多尺度的高维吉洪诺夫系统以及奇异奇摄动系统的空间对照结构.近年来,奇异摄动问题中的空间对照结构成为非常活跃又重要的研究领域.当我们处理奇异摄动问题时常常遇到空间对照结构,例如,对汽车碰撞模型里碰撞瞬间汽车的形变规律的分析,神经网络模型里神经元的传递规律以及在超导体表面所产生的边界层现象的解释等等.事实上,人们已经发现的奇异摄动问题中的多层现象,层套层现象以及非指数式衰减的边界层函数等都与空间对照结构有着密切的联系.空间对照结构是指奇异摄动方程的退化系统存在多个孤立根,而方程的解在这些不同的孤立根之间发生跳跃而产生的复杂结构.空间对照结构主要分为阶梯状空间对照结构和脉冲状空间对照结构两大类,它们在各自辅助系统的相空间中分别对应于异宿轨道和同宿轨道.其共同特点是在所讨论区间内存在一点t*,t*称为转移点(当然也可以存在多个转移点),因为在每个转移点的讨论完全一样,所以我们只讨论存在一个转移点的情况.事先t*的位置是未知的,需要在渐近解的构造过程中确定.在t*的某个小邻域内,问题的解会发生剧烈的结构变化,从一个解快速的跳到另一个解(阶梯状对照结构),或者解在t*的小邻域发生快速跳跃后又回到原来的解(脉冲状对照结构).本文把边界函数法和几何奇摄动理论相结合,针对高维吉洪诺夫系统以及奇异奇摄动系统的阶梯状空间对照结构进行研究.借助首次积分构造所需要的异宿轨道,用边界函数法构造形式渐近解；缝接边界函数的同时确定转移点t*的位置,在证明真解的存在性和形式渐近解的一致有效性时用到了缝接法,微分不等式以及k+σ交换引理等不同的方法.
全文共分为三部分,第一部分研究高维奇异摄动系统的空间对照结构,第二部分研究奇异奇摄动系统的空间对照结构,第三部分给出了对照结构在微分差分方程上的应用.
现谨将主要内容和研究结果概述如下：
第一章回顾了奇异摄动的历史背景,概述了空间对照结构的历史,发展过程及现状,引入了与本文研究内容相关的一些基本定义和引理,介绍了本文的工作和创新之处.
第二章研究了带慢变量的拟线性方程的阶梯状空间对照结构,用边界函数法构造了该问题的形式渐近解.用缝接法对轨道进行了光滑缝接,在整个区间上证明了阶梯解的存在性和形式渐近解的一致有效性.
第三章研究了具有初边值条件(快变量给定边值条件,慢变量给定初值条件)的(M+m)维吉洪诺夫系统的阶梯状空间对照结构.借助首次积分构造高维空间的异宿轨道,利用指数二分法的一些性质和Fredholm交换引理在求解高阶边界函数的同时确定了转移点t*的位置.用边界函数法构造了形式渐近解并用缝接法证明了阶梯解的存在性和形式渐近解的一致有效性.
第四章研究了具有边值条件(快慢变量均给定边值条件)的(M+m)维吉洪诺夫系统的阶梯状空间对照结构.把传统的边界函数法和几何奇摄动理论相结合,借助首次积分构造高维空间的异宿轨道,利用指数二分法的一些性质和Fredholm交换引理在求解高阶边界函数的同时确定了转移点t*的位置.用边界函数法构造了形式渐近解,用k+σ交换引理证明了阶梯解的存在性和形式渐近解的一致有效性.
第五章研究了高维奇异奇摄动系统的阶梯状空间对照结构.把传统的边界函数法和几何奇摄动理论相结合,借助首次积分构造高维空间的异宿轨道,确定转移点t*的位置.用边界函数法构造了形式渐近解,用k+σ交换引理证明了阶梯解的存在性和形式渐近解的一致有效性.
第六章研究了弱非线性微分差分方程的内部层解,利用边界函数法和分步法构造了所讨论问题的渐近解,并且指出在t=0的边界层将对t=σ的内部层产生重要的影响.用微分不等式的方法证明了解的存在性和所构造的形式渐近解的一致有效性.
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.
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