摘要
本文旨在研究平面跨临界型转点处的分支现象和带有截断项的扩展FKPP方程行波解的异宿轨道分支.近年来,利用几何奇摄动结合动力系统理论研究奇摄动系统的分支现象已得到了较大的发展.如奇摄动系统中的鸭现象,奇摄动系统中的同、异宿轨分支等.但由于奇摄动系统的特殊性,其分支理论与方法还有待进一步的发展和完善.本文运用几何奇摄动理论和动力系统中的方法研究平面奇摄动系统中的几类分支现象,并推广了前人的一些结果.
全文共分三章.谨将具体内容和研究结果概述如下：
第一章简单介绍了奇异摄动的几何理论,叙述了鸭现象和带有截断项的反应扩散方程行波解的发展过程、背景及现状,介绍了本文的工作并提出了一些待解决的问题.
第二章研究了平面跨临界型转点处的分支现象,重点关注于鸭的产生过程及松弛振荡环的消失.鸭现象的研究始于80年代对van der Pol方程的研究,它的产生一般有两种机制,一是退化系统的平衡点通过临界流形的折点,二是临界流形的自相交.鸭现象的一个重要特征是由Hopf分支出的小极限环在控制参数的指数小变化范围内其振幅发生巨大改变进而变成松弛振荡环.本章研究了平面跨临界转点处鸭解的存在性、最大鸭的参数展开式、松弛振荡环的消失以及从Hopf分支到鸭爆炸之间小极限环的结构问题.该部分内容不同于经典鸭模型,是两种鸭的产生机制的综合,有着更复杂的动力学行为.
第三章研究了带有截断项的扩展FKPP方程.带有截断项反应扩散方程的研究始于Brunet和Derrida (1997)对离散N-粒子系统的临界波速与在连续假设下得到的临界波速之间存在偏离问题的研究,截断项的引入使得方程能更好的反应原系统的粒子性.在本章中我们首先利用几何奇摄动理论结合渐近方法得到了带有截断项的扩展FKPP方程临界波速的存在性及其渐近展开式,然后利用指数二分法和Evans函数方法得到了行波解在指数加权空间中的局部渐近稳定性.
This thesis aims to investigate planar transcritical type turning point bifur-cation and heteroclinic bifurcation of travelling wave in extended Fisher/KPP equation with cut-off. For the past few years, it has gained great development in studying bifurcations of singularly perturbed systems by means of geometric singular perturbation theory combining the theory of dynamical systems. Such as canards, homoclinic and heteroclinic bifurcation in singularly perturbed systems. Because of the singularity for the singular perturbation problems, the bifurcation theory remains to be further developed and improved. In this thesis, several pla-nar singularly perturbed problems with bifurcations were investigated by means of geometric singular perturbation theory and the theory of dynamical systems, some results of predecessors were extended.
The dissertation is divided into three chapters. The main results are outlined as follows:
Chapter One introduces the history and actuality for the geometric singular perturbation theory, canard phenomenon and reaction-diffusion equations with cut-off. The work of this thesis is outlined and some remaining problems are given.
Chapter Two is devoted to investigate transcritical type turning point bifur-cation on the plane, focus on the birth of canards and the vanish of relaxation oscillations. Canard phenomenon was first found and studied in van der Pol equa-tion in the1980s. Generically, canards can emerge from two possible mechanisms, one is the equilibrium of the reduced problem passing through the fold point of the critical manifold, the other is due to the self-intersection of the critical manifold. An important feature of canard phenomenon is that the transition from small Hopf-type cycles to large relaxation oscillations occurs in an exponentially small parameter interval. The existence of canard and the disappearance of relaxation oscillation in planar transcritical type turning point were proved, asymptotic ap-proximations of the control parameters were obtained, and the configuration of small limit cycles from Hopf bifurcation to canard explosion was given by means of blow-up transformations combined with standard tools of dynamical systems theory. This content is different from the classic canard model, it is a combination of the two mechanisms with complicated dynamic quality.
Chapter Three deals with travelling wave in extended Fisher/KPP equation with cut-off. Cut-offs were first introduced by Brunet and Derrida in1997to model fluctuations in propagating fronts that arise in the large-scale (or mean-field) limit of discrete N-particle systems. They found that particle attribute of the discrete systems can be modeled by introducing a small cut-off∈on the reaction term in the deterministic reaction diffusion equations. In this chapter, we prove the existence and uniqueness of travelling wave in extended Fisher/KPP equation with cut-off, derive the asymptotic expansion of the corresponding propagation speed. Further by exponential dichotomies and Evans function method, we prove that the wave with critical speed is locally exponentially stable in some weighted spaces.
引文
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