Affine-Periodic Orbits For Discrete Dynamical Systems

Periodicity is a common phenomena in the nature, many phenomena in the real world shows a periodic behavior, therefore researches on periodic solu-tions have always been a central topic in the study of the dynamical systems. But not all the natural phenomena can be described by alone periodicity. On basis of periodic solutions, people introduced the concepts of quasi-periodic solutions, almost-periodic solutions and almost automorphic solutions to de-scribe those natural phenomena which are nearly periodic. In fact, some sys-tems often exhibit certain periodicity in time and symmetry in space. In the study of periodic solutions of continuous dynamical systems, Y Li et al. in-troduced a new concept of periodic system and named it as affine-periodic system, they proved the existence of affine-periodic solutions for continuous dynamical systems. The problem of periodicity for discrete dynamical system is also widely concerned. In the present thesis, we consider affine-periodic sys-tems, and discuss the existence of affine-periodic orbits for discrete dynamical systems.In the first chapter, we introduce some basic concepts, such as (Q, N)-affine-periodic system, and (Q,N)-affine-periodic orbit. These concepts play an important role in a later study.consider the following system xn+1-xn= f(n, xn), (0.0.3) where n∈N,f:N×Rm�'Rm is continuous with respect to xn.If there exist some N∈N+,Q∈GL(m),such that f(n+N,x)=Qf(n,Q-1x)(?)(n,x)∈N×Rm.. Then system(0.0.3)is called a(Q,N)-affine-periodic system.In the second chapter,we consider the existence of(Q,N)-affine-periodic orbit for(Q,N)-affine-periodic system,where Q∈O(m).In Section 2.1,with the aid of auxiliary system of(0.0.3) xn+1-xn=λf(n,xn), (0.0.4) where n∈N,the function f:N×Rm�'Rm is continuous with respect to xn, Q∈O(m),λ∈[0,1].We introduce an existence theorem for affine-periodic orbits of system(0.0.3).Theorem 0.0.1 Let D(?)Rm be a bounded open set.If the following conditions hold for auxiliary system(0.0.4)(H1)For eachλ∈[0,1],every possible affine-periodic orbit{xn:n∈N} of system(0.0.4)satisfies Xn(?)D,n∈N; (H2)The Brouwer degree deg(g,D∩Ker(I-Q),0)≠0,if Ker(I-Q)≠{0}, where P:Rm�'Ker(I-Q)is an orthogonal projection.Then system(0.0.3)has at least one(Q,N)-affine-periodic orbit{xn*:n∈ N),xn*∈D for all n.By constructing homotopy mapping,via the topological degree theory,we proved Theorem 0.0.1 in Section 2.1.This theorem offers a topological method to study the existence of affine-periodic orbits in theory.When dealing with specific problems,we hope to have a more directly method. Hence in Section 2.2,we proved two different results on basis of Lyapunov function method as follows by using Theorem 0.0.1.Theorem 0.0.2 Consider system(0.0.3). If there exist C1 functions Vi(x),i=0,1,2,…,l and σ>0,such that(H3)For Mi large enough,|<▽Vi(xn),f(n,xn)>|≥σ>0 (?)|xn|∈Mi,i=0,1,2,…,l,n∈N. And if Ker(I-Q)≠{0},and xn∈Ker(I-Q), |<▽Vi(xn),Pf(n,xn)>|≥σ>0 (?)xn∈Ker(I-Q);|xn|≥Mi,i=0,1,2,…,l, where P:Rn�'Ker(I-Q)is an orthogonal projection;(H4)The Hessian matrix is positive semi-define,if <▽Vi(xn),f(n,xn)>>0,the Hessian matrix is positive semi-define,if <▽Vi(xn),f(n,xn)>>0,(H5)(H6)The Brouwer degree deg(▽V0,BM0∩Ker(I-Q),0)≠0, if Ker(I-Q)≠{0}, where Bp={p∈Rm:|p|<ρ}.Then system(0.0.3)has at least one (Q,N)-affine-periodic orbit.Theorem 0.0.3 Consider system (0.0.3). Let V:D�'R be a C1 func-tion such that(H7) D(?)Rm is a bounded open set;(H$) There exists a constant σ>0 such that (▽V(xn),f(n,xn))≥σ(?)(n,xn)∈N－(?)D. And if Ker(I-Q)≠{0}, (VV(xn), Pf(n, xn))≥σ(?)(n, xn)∈N×(?)(D∩Ker(I-Q)), where P:Rm�'Ker(I-Q) is an orthogonal projection;(H9) The Brouwer degree deg(▽V(xn), D∩Ker(I-Q),0)≠0, if Ker(I-Q)≠{0}.Then system (0.0.3) has at least one (Q, N)-affine-periodic orbit{xn*:n∈ N}.Theorem 0.0.3 offers a method to prove the existence of affine-periodic orbits by Lyapunov functions.Invariant region principle is an important method to study the periodic solutions for differential equations, but it is rarely used to study the almost periodic or quasi-periodic solutions. In Section 2.3, we proved an invariant region principle in discrete affine-periodic systems. Our affine-periodic orbits might be quasi-periodic. In some sense, this theorem improved the former principle.Theorem 0.0.4 Consider system (0.0.3), where f is (Q, N)-affine-periodic. Let D C Rm be a bounded open and simply connected set, such that (?)D is piecewisely smooth. Let H(f) denote the hull of f. If the following conditions hold(H10) For every (n,p)∈N×(?)D and h∈H(f), h(n,p) is inward to D;(H11) g(a)≠0 for all a∈(?)D, where P:Rn�'Ker(I-Q) is an orthogonal projection.Then system (0.0.3) has at least one (Q, N)-affine-periodic orbit{xn*:n∈ N}.In qualitative theory of differential equation, whether a certain system has stability contains a bounded orbit is a problem which people concerns, for example, if a system has stability, whether it has a periodic orbit, quasi-periodic orbit or almost periodic orbit? In the third chapter, we prove a more intuitive existence of asymptotic stability affine-periodic orbit for discrete (Q, N)-affine-periodic system via fixed point theory, where Q∈O(m).Theorem 0.0.5 Consider system (0.0.3), Let a:N�'R+\{0} satisfies r=limk�'∞akN<1.If the following conditions hold{H12) There exists an orbit{zn:n ∈N} of (0.0.3);(H13) For any two orbits{xn,x0:n∈N},{yn,y0:n∈N}, which satisfy |xn,x0-ynmy0|≤an|x0-y0|,n∈N.Then system (0.0.3) has a unique asymptotic stability (Q, N)-affine-periodic orbit.Theorem 0.0.6 Consider system (0.0.3). If system (0.0.3) is asymptoti-cally stable, then system (0.0.3) has a unique asymptotic stability (Q, N)-affine-periodic orbit.As important systems, dissipative systems are widely found in nature. The study of periodic orbits for affine dissipation systems has significance in theory and application. In the fourth chapter, we study the existence of affine-periodic orbits for discrete (Q,N)-affine-dissipative system, where Q∈ GL(m).Theorem 0.0.7 If system (0.0.3) is affine-dissipative system, then sys-tem (0.0.3) has at least one (Q,N)-affine-periodic orbit.