【作者】 赵冬华；

【导师】 阮炯；

【作者基本信息】 复旦大学 ， 应用数学， 2005， 博士

【摘要】 本文主要研究非线性动力学中两个方面的问题。第一部分为三类特殊类型的时滞微分方程解的稳定性、周期性以及振动性分析。其中第一类为分段连续的时滞微分方程,简称为EPCA。常微分方程的数值求解可以自然的产生EPCA,时滞微分方程的数值求解也可以产生EPCA。第二类为时滞依赖于状态的微分方程。虽然时滞依赖于状态的方式我们知之甚少,但这却是客观存在的,生物学家对于海豚集体自杀行为的解释提供了最好的例证。第三类为具有两个不同时滞的微分方程,即双时滞微分系统。 第二部分为混沌和分形应用的三个实例讨论。从上个世纪六七十年代开始,混沌和分形在很多领域都有了广泛而深刻的应用,本文对三个不同问题进行了讨论。第一个问题是关于常微分方程离散化过程可能引起的复杂行为的研究,由于误差的客观存在,可能出现bubble现象——双峰映射通向混沌的道路。第二个问题是关于混沌时间序列预测方法的研究,提出了新的联合预测方法,使得在预测时可以提供关于时间序列更多的信息。第三个问题是关于分形粗糙面散射的研究,反演得到分形粗糙面的分维数,实现粗糙面重构。 本文的第一章,介绍了时滞微分方程的研究进展,特别的关于分段连续的时滞微分方程,时滞依赖于状态的微分方程以及具有两个时滞的微分方程的产生背景和模型,阐述了这几类时滞微分方程理论研究的必要性;介绍了混沌及分形的发展,阐述了研究混沌和分形的重大意义及本文所讨论实例的背景和模型;同时也给出了本文的结构。 本文的第二章介绍了一类推广的捕捞模型,该模型属于分段连续时滞的微分方程。研究了其正平衡解的全局吸引性,和可能出现的复杂动力学行为——混沌;利用概周期序列的概念研究了分段连续时滞的微分方程的概周期解的存在性。更多还原

【Abstract】 In this thesis, we main study two kinds of questions of nonlinear dynamics. The first part is about the stability, periodicity and oscillation character of the solution of three kinds of special delay differential equations. One of them is piecewise continuous argument delay differential equation, simply called EPCA. Numerical solutions of ordinary equation can naturally arise EPCA. At the same time, numerical solutions of delay differential equation can also give birth to EPCA. The second is about delay dependent differential equation. Although we know little about the natural way of delay dependent, but it is the objective reality. Biologists’ explanation about the behavior of dolphins who take suicide near shallow sea must be the best illustration.The second part is about the application of chaos and fractal. From sixty or seventy age of last century, chaos and fractals have been widely applied in many fields, we study of it from three aspects. Discreting ordinary equations may bring complex behavior-bubbling and bistable phenomena, discussed the route to chaos of bimodal map. About the important application of chaos, chaotic time series’ predict, we present a new method, called combined predict method, which can provide more information about the series. Fractals has the great advantage at describing natural objects, and scattering from complex environment has been the focus of research, study of scattering from fractal rough surface is our additional work.In Chapter 1, we introduce the research progress for delay differential equations in recent years, especially about the backgrounds and models of piecewise continuous argument delay, delay-dependent and two delays differ-ential equations. We show that it is necessary to analyze these kinds of models. At the same time, we provide the structure of the thesis.In chapter 2, we introduce a generalized harvesting model, which is belong to piecewise continuous argument delay equations, we present sufficient condition to ensure the global attractivity of unique positive equilibrium and explore the possibility of emerging complex behavior-chaos. We also analyze the existing of almost periodic solution by the notion of almost periodic sequences.In chapter 3, we introduce the population model of delay-dependent, We present sufficient condition to guarantee the global attractivity of the unique positive equilibrium, and estimate the oscillation property of its solution. Taking advantage of coincidence degree theory we analyze the existing of periodic solution.In chapter 4 we analyze the oscillation property of solution in two delays differential equations.In chapter 5,we discuss different behavior under three simple discreting methods, and explore bubbling phenomena which exists in many bifurcation diagrams.In chapter 6, we discuss the predict method of chaotic time series. There are abundant dynamical information in chaotic time series, how to extract the information and apply them in reality is an important application. We present a new combined predict method, which is based on one order weighted local predict method, combined interval estimate method, we can give predict result and confidence interval, it is complement of point predict method.In chapter 7, we study scattering from fractal rough surface. We apply Monte Carlo method, take a band-limited Weierstrass-Mandelbrot function to model fractal rough surface, present minimal object functions to inverse fractal更多还原