Dynamical Behavior Analysis Of Special Delay Differential Equations And The Application Of Chaos And Fractals

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...