Study On Some Problems Of Chaos In Discrete Dynamical Systems

Chaos is one of the important contents of nonlinear science, the inherent properties of nonlinear dynamical systems, and a common phenomenon of non-linear systems. The subject for studying chaos is Chaology. In general, chaos means a random-like behavior (intrinsic randomness) of deterministic systems without adding 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 are unpredictable.Chaos is a new cross subject, which develops with the rapid development of modern technology, especially based on the occurrence and wide applications of computer technology. An important speciality of chaos is that it breaks down the boundaries among different fields. The concepts and methods of universal-ity, power law, self-similarity, fractal, strange attractors, renormalization group of chaology has gone beyond the background of mathematics and physics into chemistry, biology, geonomy, medicine and social science. The famous physi-cist, J. Ford, claimed that chaos is the third revolution in physics after quantum mechanics and relativity theory in the 20th century.Although chaos has been studied by many scientists for nearly half a century, there are still no widely accepted mathematical definitions. On one hand, it is because chaotic systems are indeed very complicated and there are different understandings from different viewpoints; on the other hand, Chaoiogy is a cross subject, and the knowledge of different fields for chaos is different. The usually used definitions in mathematics are the following:chaos in the sense of Li-Yorke [43], Devaney [24], and Wiggins [92], distribution chaos for discrete dynamical systems [97]; and chaos in the sense of Smale:s horseshoe for diffeomorphisms [34,79,96].A major topic in the study of chaos is criteria of chaos. There are many elegant existing results. For example, for continuous interval maps there is the well-known result "periodic 3 implies chaos" given by Li and Yorke [43], and further it has been proved that nonpower of 2 period, turbulence, and positive topological entropy all imply chaos in the sense of Li-Yorke [4]. For higher-dimensional maps, there are the famous Marotto theorem [52,53.69], the snap-back repeller theory under weaker conditions [75], and the heteroclinical repellers theory established by Lin and Chen [48]. For maps in general Banach spaces and complete metric spaces, there are the generalized snap-back repeller theory and the coupled-expansion theory established by Shi with her coauthors [68, 76], and the heteroclinic cycles connecting repellers theory established by Li and Shi [46], etc. For diffeomorphisms, there is the well-known Smale horseshoe for two-dimensional spaces, and the Smale-Birkhoff homoclinic theorem in higher-dimensional spaces [79]; K. Shiraiwa and M. Kurata generalized Smale-Birkhoff theorem to maps on higher-dimensional manifolds, for which the maps may not be diffeomorphisms [77]; H. Steinlein and H. Walther got the result that transversal homoclinic orbits for a C1 map on a Banach space implies chaos in the sense of Devaney [81]. Further, J. K. Hale and X. Lin generalized the theorem to Ck(k≥0) maps on Banach spaces and the map may not be a diffeomorphism [28]. Y. Yan and M. Qian proved that a transversal heteroclinic cycle implies a transversal homoclinic orbit on an n-dimensional manifold [95], hence, it is chaotic. K. Burns and H. Weiss showed that a transversal homoclinic orbit or transversal heteroclinic cycle implies a horseshoe by using a geometric method [8]. W. Li proved that a transversal homoclinic orbit to a saddle-focus fixed point of a diffeomorphism in two dimensional spaces implies a horseshoe [44]. B. Deng studied the existence of a transversal homoclinic orbit to a saddle-focus fixed point for a diffeomorphism in higher dimensional spaces implies a horseshoe [23]. The usually used criteria of chaos in physics and engineering are that solutions a system are bounded and a system has positive Lyapunov exponents or positive topological entropy.Since Li and Yorke gave the definition of chaos, the study on this chaotic phenomenon has attracted many scientists. Many scholars came to study prop-erties of the set which consists of maps that are chaotic in the sense of Li-Yorke, such as the distribution of the set in some map spaces. Kloeden showed that the set of continuous interval maps, which have periodic points of period 3 and then are chaotic in the sense of Li-Yorke, is dense in C(I,I) [37]. Butler and Pianigiani further proved that the set of continuous interval maps, which have periodic points of period not equal to powers of 2 and then are chaotic in the sense of Li-Yorke, contains a dense open set in C(I, I) [9]. Later, applying the result in [39] about existence of nontrivial ergodic invariant measure, Siegberg proved that for each integer k≥2, there is a dense open set Ak (?) C(I, I) such that every f∈Ak satisfies the following properties:(i)f is chaotic in the sense of Li-Yorke;(ii) the topological entropy h(f)> log k;(iii) there exists a continuous invariant measure with respect to f. Consequently, there is a dense residual set R(?)C(I, I) such that every f∈R satisfies the above properties (i) and (iii), and moreover h(f) =∞[78]. By the result of [3], a continuous interval map f is chaotic in the sense of Li-Yorke if it has a positive topological entropy. And hence, it is also chaotic in the sense of Devaney.For higher-dimensional and infinite-dimensional cases, Siegberg gave the fol-lowing partial n-dimensional analogue:if (P,τ) is a compact polyhedron which is acyclic in Rn, then there is a residual set R(?) C(P,P) such that every f∈R is almost chaotic in the sense of Li-Yorke [78, Theorem 3.5]. In the infinite-dimensional case, he applied a finite-dimensional approximation method and showed that a compact continuous map space contains a dense set of maps that are chaotic in the sense of Li-Yorke [78, Theorem 2.5]. Recently, Mimna ob-tained that in the case thatΩis a compact n-cube in Rn (the Cartesian product of n compact intervals), there exists a dense open set W in C(Ω,Ω) such that every map in W is not topological transitive inΩ, and hence not chaotic in the sense of Devaney on the whole domainΩ[56, Theorem 2].Meanwhile, in the 1960’s, Smale [80] studied density of hyperbolicity. Some scholars believed that hyperbolic systems are dense in spaces of all dimensions, but it was shown that the conjecture is false in the late 1960’s for diffeomor-phisms on manifolds of dimension> 2. The problem whether hyperbolic systems are dense in the one-dimension case was studied by many scholars. It was solved in the C1 topology by Jakobson [33], a partial solution was given in the C2 topol-ogy by Blokh and Misiurewicz [5], and C2 density was finally proved by Shen [67]. In 2007, Kozlovski, Shen and Strien got the result in Ck topology; that is, hyper-bolic (i.e. Axiom A) maps are dense in the space of Ck maps defined in a compact interval or circle, k= 1,2,…,∞,ω[38]. At the same time, some other scholars considered the distribution of hyperbolic diffeomorphisms in Diff[M], where M is a manifold. Just like the work of Smale. Palis [59,60] gave the following conjec-ture:(1) any f∈Diff(M) can be approximated by a hyperbolic diffeomorphism or by a diffeomorphism exhibiting a homoclinic bifurcation(tangency or cycle), (2) any diffeomorphism can be CT approximated by a Morse-Smale one or by one exhibiting transversal homoclinic orbit. Later, Pujals and Sambarina showed that the conjecture (1) holds for C1 diffeomorphisms of surfaces [63]. And some good results have been obtained, such as any diffeomorphism can be C1 approximated by a Morse-Smale one or by one displaying a transversal homoclinic orbit [18], any diffeomorphism can be C1 approximated by one that exhibits either a homoclinic tangency or a heterodimensional cycle or by one that is essentially hyperbolic [19].For control of chaos, many efficient controlling methods have been found to eliminate chaos when it is harmful, such as the OGY method, the occasional pro-portion feedback technique (OPF technique), the pulse controlling method, the sliding modular controlling method, the linear and nonlinear controlling method, the self-feedback controlling method, etc. When chaos is useful, some researchers also have found many methods to create chaos. For discrete systems, it has been made a great progress in anti-control of chaos. Chen and Lai firstly proposed a method of state feedback control for chaotification for any finite-dimensional discrete systems. Shi with her coauthors extended this method to general Banach spaces. For the history and more detailed developments about the anti-control of chaos for discrete systems, we refer to [13,14,25,30,89,101].In this dissertation, we mainly study the existence of transversal homoclinic orbits for continuous maps in finite-dimensional spaces, the density distribution of chaotic maps in some map spaces in Banach spaces, and the chaotification for discrete dynamical systems in finite-dimensional spaces. It consists of four chapters and the main contents are as follows.In Chapter 1, we summarize the development of chaos, and give some pre-liminaries, including several definitions of chaos that are often used and some other basic concepts in discrete dynamical systems, some fundamental theory for symbolic dynamical system, and the definition of ttransversal homoclinic orbits for continuous maps in Banach spaces. We also recall some existing criteria of chaos in dynamical systems.In Chapter 2, we consider a continuous map in an n—dimensional space, and give a sufficient condition for the existence of transversal homoclinic orbits. We first give the definition of transversal heteroclinic cycle for continuous maps on Banach spaces, then generalize the result that a transversal heteroclinic cycle implies a transversal homoclinic orbit for a diffeomorphism in a finite dimensional space to continuous maps. Finally, we give a sufficient condition for the existence of transversal homoclinic orbits for continuous maps in finite dimensional spaces. Furthermore, we present two examples and their computer simulations for illus-tration. In particular, we do not require that the map has a homoclinic orbit.In Chapter 3, we mainly consider the density of chaotic maps in continuous map spaces, including coupled-expanding maps, maps with snap-back repellers, maps with transversal homoclinic orbits. First, we give the map space that we study. In the map space, we construct a class of coupled-expanding maps, maps with snap-back repellers, and maps with transversal homoclinic orbits to illustrate the density of the maps. Simultaneously, we discuss the topological entropy for finite-dimensional spaces.In Chapter 4, we mainly study chaotification of discrete dynamical systems in finite-dimensional spaces. By applying the idea for constructing transversal homoclinic orbits in Chapter 3, we establish a new chaotification scheme. Finally, we provide an example with computer simulations for illustration.