Chaos In Periodic Discrete Systems

Chaos science is a cross subject.As one of the focuses of nonlinear science, it has intersected with mathematics,physics,chemistry,biology,physiology,psychology, electronics,information science,social science and many other fields.It has been applied extensively to mechanical engiueering,chemical engineering,biological engineering,medical engineering,electronic engineering,information engineering, etc..Chaos has been studied for a long time.In 1903,French mathematician and physician H.Poincare found chaos when he was studying three-body problem. Maybe he is the first scholar to know the existence of chaos in the world.In 1954,Soviet mathematician A.N.Kolmogorov found that conditional period kept unchanged when a Hamiltonian function has a small change.In 1963,his student V. I.Arnold gave a rigorous mathematical proof of his result.And at almost the same time,Swiss mathematician J.Morser proposed an improved proof.The KAM Theory, which was named by the first letters of names of these three mathematicians, has become the most successful result in the research of unintegrable systems for more than one century.In 1963,American meteorologist Lorenz found nonperiodic fluid:a deterministic system might have nonperiodic disordered behavior,i.e.,the result has sensitive dependence on initial conditions,which is just the interesting Butterfly Effect.It is usually considered as the essence of chaos.Lorenz's paper Deterministic nonperiodic flow[1]and his other three papers published later became an important break-through in the research of chaos.In 1975,T.Li and J.Yorke studied discrete dynamical systems induced by one-dimensional continuous maps and obtained a famous result:Period 3 implies chaos.They firstly introduced the concept of chaos to science[2].Henceforth,as a new scientific term,chaos appears formally in scientific references.According to different requirements in the research of different problems,there appear several different definitions of chaos.At present, the following three definitions of chaos are often used:chaos in the sense of Li-Yorke [2];chaos in the sense of Devaney[3]and chaos in the sense of Wiggins[4].Criteria of chaos in autonomous discrete systems have been extensively studied for many years with many significant results up till now.For one-dimensional autonomous discrete systems induced by interval maps,the famous results:period 3 implies chaos[2];non power of 2 period,turbulence and topological entropy all imply chaos in the sense of Devaney and Li-Yorke[5].For higher-dimensional autonomous discrete systems,F.R.Marotto established a criterion of chaos which have snap-back repellers on R~n[6];Y.Shi and G.Chen gave its improvement version and weakened its conditions[7];W.Lin and G.Chen established heteroclinical repellers theorem[8].For infinite-dimensional autonomous discrete systems,Shi et al.pro-posed snap-back repellers theory and coupled-expanding theory in general Banach spaces and complete metric spaces[7,9-14],and Li et al.established heteroclinic cycles connecting repellers theorem[15].In the real world,many mathematical models are described by time-varying discrete systems.But they are simplified as autonomous discrete systems for convenience. The essential difference between a time-varying discrete system and a autonomous discrete system is that the former is induced by a family of maps. Dynamical behaviors of time-varying discrete systems are much more complicated than those of autonomous discrete systems.For example,a finite-dimensional linear autonomous discrete system can not be chaotic,but a finite-dimensional linear time-varying discrete system may be chaotic in the sense of Li-Yorke[16,Examples 2.1,and 2.2].Hence,it is more difficult to study chaos in time-varying discrete systems than that in autonomous discrete systems.Thus,there are a few results about chaos in time-varying discrete systems.To the best of our knowledge,only references[16,17]discussed some chaotic problems in time-varying discrete systems, where[17]studied chaos of time-varying discrete systems induced by a sequence of maps in a metric space under iterative way and successive way,respectively,and obtained that chaos in the iterative way does not imply chaos in the successive way, and vice versa;and[16]generalized the concept of chaos in autonomous discrete systems to time-varying discrete systems and established criteria of chaos in the sense of Li-Yorke for finite-dimensional linear time-varying discrete systems and a criterion of chaos in the strong sense of Li-Yorke for general time-varying discrete systems.In the current literature,there are few results about chaos in periodic discrete systems,which is a special kind of time-varying discrete systems.Reference [18]gave an example to illustrate that the composition of two chaotic maps may not be chaotic.In fact,the two chaotic maps induce a 2-periodic discrete system.Based on the work of[16],we establish two criteria of chaos and study small perturbation problems of periodic discrete systems.This paper consists of two chapters.In Chapter 1,we obtain some basic conclusions, which are necessary preparations for establishing criteria of chaos and studying small perturbation problems of periodic discrete systems in Chapter 2.The main contents are introduced as follows:In Chapter 1,we mainly discuss the relationship between dynamic behaviors of periodic discrete system and its induced autonomous discrete system.We obtain that chaos in the sense of Devaney(Wiggins) and in the(strong) sense of Li-Yorke in a autonomous discrete system imply chaos in the sense of Devaney(Wiggins) and in the(strong) sense of Li-Yorke in its original periodic discrete system,respectively; and under certain conditions,chaos in the sense of Devaney(Wiggins) and in the (strong) sense of Li-Yorke in a periodic discrete system imply chaos in the sense of Devaney(Wiggins) and in the(strong) sense of Li-Yorke in its induced autonomous discrete system,respectively.In particular,we show that finite-dimensional linear periodic discrete systems can not be chaotic in the sense of Li-Yorke.In Chapter 2,we study criteria of chaos and small perturbation problems of periodic discrete systems.Based on the coupled-expansion theory of autonomous discrete systems,relevant results in[16]and conclusions obtained in Chapter 1,we establish two criteria of chaos in the sense of Li-Yorke,or in the sense of Devaney and strong sense of Li-Yorke for periodic discrete systems induced by coupled-expanding maps.Further,we study invariance of some chaotic behaviors of chaotic periodic discrete systems perturbed by periodic families of maps in Euclidean spaces.