几类非光滑动力系统的研究

发布时间：2018-03-20 18:26

本文选题：奇异摄动　切入点：脉冲微分方程　出处：《华东师范大学》2016年博士论文　论文类型：学位论文

【摘要】：本文针对几类非光滑动力系统进行了研究.主要包括：一类脉冲微分方程的多尺度研究；几类非光滑奇摄动方程的空间对照结构的研究.本文的主要内容分为以下几章.第一章为绪论.主要回顾了奇摄动问题和非光滑动力系统的历史背景和发展过程,引入了与本文研究内容相关的一些概念和定理,同时也介绍了本文的工作及创新之处.第二、三章对一类脉冲微分方程进行多尺度的研究.脉冲微分方程充分考虑到瞬时突变对状态的影响,其解往往是分段连续的,解的这种特性给脉冲微分方程的研究带来了困难与不便.为此,利用奇摄动的思想,通过引入适当的奇摄动项,这两章分别将原脉冲微分方程扩充成具有无穷大初值的奇摄动问题和临界奇摄动边值问题,构造了相应奇摄动问题的连续/光滑的多尺度渐近解来有效地刻画原脉冲微分方程的不连续解,同时也论证了渐近解的一致有效性,从而为脉冲微分方程的研究提供了一种新的途径.第四章研究了非光滑的二阶半线性奇摄动方程的空间对照结构.在区间内的某点t处改变方程的右端函数,从而导致了附加方程平衡点类型的变化,使得左区间可能出现脉冲状的空间对照结构解,右区间可能出现阶梯状的空间对照结构解.通过相平面,分析了整个区间上解存在的可能类型.利用边界层函数法,构造出左区间脉冲状解和右区间阶梯状解的渐近表达式,同时也确定产生脉冲状解和阶梯状解的点的渐近表达式.而在t点处：运用缝接法,对解进行光滑缝接.最后证明了整个区间上解的存在性和渐近解的一致有效性.第五章研究了带慢变量的非光滑拟线性奇摄动方程组的空间对照结构.利用边界层函数法构造了该问题的形式渐近解,用缝接法对解轨道进行光滑缝接,在整个区间上证明了形式渐近解的存在性和一致有效性.第六章研究了具有快慢层的非光滑奇摄动方程的空间对照结构.不同于前面的奇摄动方程的边界层具有相同的类型(尺度变化一样),本章所考虑的奇摄动方程的边界层具有不同的类型,即出现快慢层.主要包括：快慢层出现在不同端点处和快慢层出现在同端点处.在快慢层处,通过引入不同的尺度变化,分不同区间构造了问题的形式渐近解.在整个区间上,利用缝接法,得到了解的存在性和渐近表达式,同时也论证了渐近解的一致有效性并进行了余项估计.最后,总结本文的工作,并提出下一步的研究计划.
[Abstract]:In this paper, several kinds of nonsmooth dynamical systems are studied, including: the multi-scale study of a class of impulsive differential equations; The main contents of this paper are divided into the following chapters. The first chapter is an introduction. The historical background and development process of singularly perturbed problems and nonsmooth dynamical systems are reviewed. Some concepts and theorems related to the contents of this paper are introduced, and the work and innovations of this paper are also introduced. In chapter three, we study a class of impulsive differential equations with multiple scales. The impulsive differential equations take into full account the influence of transient mutations on the state, and their solutions are usually piecewise continuous. This characteristic of solution brings difficulties and inconvenience to the study of impulsive differential equations. Therefore, by introducing proper singularly perturbed terms, using the idea of singularly perturbed, In these two chapters, the original impulsive differential equations are extended to singular perturbation problems with infinite initial values and critical singularly perturbed boundary value problems, respectively. The continuous / smooth multi-scale asymptotic solutions of the corresponding singularly perturbed problems are constructed to characterize the discontinuous solutions of the original impulsive differential equations effectively, and the uniform validity of the asymptotic solutions is also proved. In chapter 4th, the spatially controlled structure of nonsmooth second order semilinear singularly perturbed equations is studied. At some point in the interval, the function of the right end of the equation is changed. This result in the change of equilibrium point type of the additional equation, which makes it possible for the left interval to have a pulse-shaped spatial contrast structure solution, and the right interval for a step space control structure solution. The possible types of solutions on the whole interval are analyzed. By using the boundary layer function method, the asymptotic expressions of the left interval impulsive solutions and the right interval step solutions are constructed. The asymptotic expressions of the points that generate the impulsive solution and the step solution are also determined. The existence of solutions over the whole interval and the uniform validity of asymptotic solutions are proved. In Chapter 5th, the spatially controlled structures of nonsmooth quasilinear singularly perturbed equations with slow variables are studied. The formal asymptotic solution of the problem is constructed by the layer function method. The slit method is used to smooth the slit of the solution track. The existence and uniform validity of formal asymptotic solutions are proved in the whole interval. In chapter 6th, we study the spatially controlled structure of the nonsmooth singularly perturbed equation with fast and slow layers. The boundary layer of the singular perturbation equation is different from that of the previous singularly perturbed equation. The boundary layer of the singularly perturbed equation considered in this chapter has different types. That is, the fast and slow layers appear at different endpoints and the fast and slow layers appear at the same endpoints. At the fast and slow layers, by introducing different scale changes, the asymptotic solutions of the problem are constructed in different intervals. The existence and asymptotic expression of the solution are obtained by using the slit method, and the uniform validity of the asymptotic solution is also proved and the remainder estimates are given. Finally, the work of this paper is summarized and the next research plan is put forward.
【学位授予单位】：华东师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O19

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