Chaos In Partial Differential Equations And Integral Systems

Chaos is one of central topics research on the nonlinear science and is a universal dynamical behavior of nonlinear dynamical systems.Meanwhile, it has a global and essential effect on the development of nonlinear dynamics, and some original research works of nonlinear dynamics were connected with chaos.However,before the end of 1950's and the establishment of chaos theory,the concept of chaos was very ambiguous.Even now,there are different understandings of chaos in different fields.In general,chaos means a random-like behavior(intrinsic randomness)in deterministic systems without any stochastic factors.The most important characteristic of a chaotic system is that its evolution has highly sensitive dependence on initial conditions.So, from a long-term perspective,future behaviors of a chaotic system are unpredictable.Research on chaos in dynamical systems has attracted a lot of attention from many scientists and mathematicians.In 1975,Li and Yorke[1]invesrigated a continuous map on a interval and obtained the well-known result: "period 3 implies chaos".This criterion plays an important rule in studying chaos problems of one-dimensional discrete dynamical systems.They are the first ones who introduced a mathematical definition of chaos.Later,there appeared several different definitions of chaos[2-5];some are stronger and some are weaker,depending on requirements in studying different problems. In 1978,F.R.Marotto was inspired by the work of Li-Yorke,generalized the Li-Yorke's theorem to n-dimensional real space,and introduced the concepts of expanding fixed point and snap-back repeller.The Marotto's theorem[6, Theorem 3.1]proved that a snap-back repeller implies chaos in the sense of Li-Yorke.Recently,Y.Shi and G.Chen captured the essential meanings of the expanding fixed point and snap-back repeller,and generalized these two concepts for the continuously differentiable maps in Rⁿ to maps in general metric spaces.And they established several criteria of chaos in discrete dynamical systems on complete metric spaces[7,8].Many partial differential equations and intergral equations,which are deduced by models of dynamical systems, are often studied on Banach spaces.Nonlinear vibrations in mechanical and electronic systems are always an important topic studied by scientists and engineers[9].In recent years,some study on this problem was focused on chaotic phenomena.But it seems that there are a little results obtained in the mathematical study of chaotic vibrations in mechanical systems governed by partial differential equations(PDEs) containing nonlinearities.In general,more mathematical knowledges and techniques are required to study dynamical behaviors of PDEs.PDEs have many different types.Here,we will only study chaotic vibrations of a vibrating string with nonlinear boundary conditions in Chapter 2.Integral equations are an important branch of the advanced mathematics. It has very important applications to mechanics,mathematical physics, engineering,etc.Moreover,it is well-known that the integral equation can be iterated to find its numerical solution.But in some cases there may be some strange phenomenon that a small change on the initial value may result in a great difference in approximate solution,even the graph of the solution may be complicated,which is now called a chaotic phenomenon.In Chapter 3,we will establish several criteria of chaos for integral systems.These results may give an explanation for the above mentioned problem.In this dissertation,we mainly study chaos problems on vibrations of the one-dimensional wave equation and small perturbation of a class of chaotic discrete systems on Banach spaces.This dissertation consists of three chapters. Their main contents are briefly introduced as follows.In Chapter 1,we summarize the development of chaos and give some preliminaries,including several basic definitions of chaos that often used in mathematics,some other concepts in discrete dynamical systems,and several criteria of chaos.In Chapter 2,we study the chaotic vibration of a one-dimensional wave equation.The wave equation is an important type of the partial differential equations in the study of engineering technology and mechanics.In this chapter, we consider the one-dimensional vibrating string satisfying w_tt-w_xx=0. x∈(0,1).It is an infinite-dimensional harmonic oscillator with the boundary conditions:at.the left end x=0,it satisfies a linear condition,while at the right,end x=1,it satisfies a nonlinear boundary condition.First,the problem is reformulated into an equivalent first-order hyperbolic system.By the method of characteristics,the problem is reduced to a discrete iteration problem:u_n+1=H₀(u_n),where H₀ is an interval map.This PDE problem is said to be chaotic if the map H₀ is chaotic.In addition,we provide an example with computer simulations for illustration.In Chapter 3,we study a small perturbation of a class of chaotic discrete systems on Banach spaces.First,we discuss the small perturbation of a class of chaotic discrete systems.In general,a chaotic system is not structurally stable;that is,a chaotic system with a small perturbation may be chaotic or not.Here,we discuss a system is still chaotic by a small perturbation,where the system has a regular nondegenerate snap-back repeller.Second,we emply the obtaiu results to study chaotic problems of three classes of integral systems on Banach spaces and establish some criteria of chaos.