Existence Of Homoclinic And Heteroclinic Orbits Of Several Types Of Second-order Discrete Hamiltonian Systems

The research of homoclinic orbits of Hamiltonian systems dates back to Poincare’s work on celestial mechanics [51], in which Poincare discovered that the dynamical behavior of the system is very complicated and there exist infinitely many homoclinic orbits if the stable and unstable manifolds intersect transversally. Later, the work of Smale and Birkhoff showed that a system with a transversal homoclinic orbit is chaotic [62]. It is a meaningful question to determine whether or not a system has a transversal homoclinic orbit. The well known Melnikov method answers a part of this problem. Generally speaking, to investigate the existence of transversal homoclinic orbit, we need to show the existence of the homoclinic orbit and show this orbit is transversal. Hence, it is a very important step to show the existence of homoclinic or heteroclinic orbits.After the pioneering work of Rabinowitz [53], studying the existence of periodic solutions of a class of first order continuous Hamiltonian systems by the variational methods, more and more researchers found that the variational methods play an important role in the investigation of the existence of periodic, homoclinic and heteroclinic orbits of continuous Hamiltonian systems because of the variational structure. The research on the discrete Hamiltonian system is just beginning, there are lots of unsolved problems in this area.To investigate the existence of homoclinic or heteroclnic orbits via the variational method, one usually needs to construct a suitable variational func-tional on a proper Hilbert space such that a non-zero critical point of the functional is a non-trivial homoclinic or heteroclinic orbit of the system. So, we convert this problem to the study of the existence of the non-zero critical points of the functional. And, the variational approach is very effective in find-ing the critical points. We refer to [54,70] for more materials on variational methods.A great progress has been made in the research of the existence of ho-moclinic orbits of Hamiltonian systems. In [56], Rabinowitz obtained the ex-istence of a homoclinic orbit of a kind of second-order differential equations by using a series of periodic orbits to approximate the homoclinic orbit. In [20], Coti-Zelati and Rabinowitz investigated the existence of homoclinic or-bits of second-order Hamiltonian systems with the superquadratic potential assumptions at both the origin and at the infinity. In [65], Sere introduced the multibump solutions and depicted the complicated dynamical behavior combined with the Bernoulli shift. In [27], Ding applied the spectral theo-ry to investigate several types of supquadratic or subquadratic second-order differential equations.There are lots of good results that have been obtained in the study of the existence of heteroclinic orbits. In [30], Felmer investigated the existence of heteroclinic orbits of a kind of one-order Hamiltonian systems with the as-sumption that the Hamiltonian is periodic with respect to one space variable and superlinear in another one. In [9], Bertotti and Montecchiari studied the existence of infinitely many heteroclinic solutions connecting degenerate equi-libria for a kind of second-order almost periodic systems. In [13], Caldiroli and Jeanjean obtained that there is a heteroclinic orbit connecting the origin and a minimal non-contractible periodic orbit, which is the limit of a sequence of homoclinic orbits with special properties. In [55], Rabinowitz studied the exis-tence of periodic and heteroclinic orbits of a class of second-order Hamiltonian systems. There are also some works on the existence of heteroclinic orbits for the pendulum equation [60]. Coti-Zelati and Rabinowitz investigated the ex-istence of heteroclinic orbits which connect two critical points of the potential function at different energy levels [21].For Hamiltonian system, although the variational method is very powerful, it is difficult to show the transversality of the homoclinic orbit. A kind of relatively weaker solution "multibump solution" is introduced to study the complicated dynamics of Hamiltonian systems. For the multibump solutions of certain systems, it is Sere’s who firstly investigated the multibump solutions of a kind of first-order Hamiltonian systems under certain conditions [65]. Later, similar construction of the orbits is obtained in different situations. For example, the degenerate case [59], the damped systems [10], the potential changing sign case [14]. However, the construction of multibump solutions in difference equations is just beginning, there is few work about the existence of multibump solutions of homoclinic orbits of difference equations.The procedure to show the existence of multibump solutions is to first use a variational argument, minimax method, to find a special family of non-trivial homoclinic solutions which can be regarded as the "one-bump" solution-s. Secondly, variational arguments are applied to get multibump solutions, i.e. solutions near sums of sufficiently separated translates of the "one-bump" so-lutions. A key assumption for the existence of multibump solutions is that the critical points of the corresponding functional are isolated. This hypothesis is used to replace the classical transversality conditions.For the research of dynamical systems, the method of modeling and sim-ulating of a real system plays an important role. Since the continuous systems can not be directly applied in the real computation, we need to transform the continuous equations into their corresponding difference equations such that we could observe the dynamical behavior of the systems through simulation-s. Hence, the research of difference equations gradually becomes important. Discrete Hamiltonian systems can be applied in many different areas, such as physics, chemistry, engineering and so on. Please refer to [1] for more infor-mation on discrete Hamiltonian systems. There are many good results on the existence of periodic, homoclinic, and heteroclinic orbits of discrete Hamilto-nian systems [33,35,41,45,46,86,88].In [33], Guo and Yu investigated the existence of periodic and subhar-monic solutions of the scalar second-order difference equation as follows: Δ2x(t-1)+f(t,x(t))=0. Lots of researchers have studied the existence of periodic, homoclinic, and heteroclinic orbits of the following type of difference equations Δ(p(t)Ax(t-1))-L(t)x(t)=f(t,x(t)), x(t)∈Rn, t∈Z, which can be written as equivalent discrete Hamiltonian systems through a proper transformation. In [46], Ma and Guo proved the existence of homoclin-ic orbits of difference equations with n=1under the periodicity assumptions on p(t), L(t), and f(t,x) in t. In [45], Ma and Guo studied the existence of homoclinic orbits of difference equation provided that f(t,x) grows superlin-early both at origin and at infinity or f(t,x) is an odd function with respect to x, where the assumption of periodicity on p(t) and L(t) are not required. In [41], Lin and Tang obtained that there exist infinitely many homoclinic orbits of the equation with the assumption that L(t) is positive definite for any t∈Z and more general conditions on f(t,z).The research of the existence of heteroclinic orbits of discrete systems is just beginning. The existence of heteroclinic orbits can be shown by studying the existence of minimizing sequences on certain function spaces. In [79], Xiao and Yu investigated the existence of heteroclinic orbits of the following discrete pendulum equation A2x(t-1)+a sin(x(t))=0, where a∈R is a parameter, x(t)∈R, and t∈Z. In [80,88], Xiao et al. and Zhang and Li studied the existence of heteroclinic orbits of the following difference equation Δ2x(t-1)+V’x(x(t))=0, where x(t)∈Rn and t∈Z.This thesis deals with three basic problems:the first is to investigate the existence of homoclinic orbits of two types of second-order discrete Hamiltonian systems; the second is to study the existence of multibump solutions of a class of second-order discrete Hamiltonian systems; the third is to look for the heteroclinic orbits of a kind of second-order discrete Hamiltonian systems.In Chapter1, some basic concepts and useful results about variational methods are introduced, and some knowledge on the spectral theory of linear operators is given.In Chapter2, we discuss the following type of second-order discrete Hamil-tonian systems: Δ2x(t-1)-L(t)x(t)+V’x(t,x(t))＝0,t∈Z,(*) where Δx(t-1)=x(t)-x(t-1), Δ2x(t-1)=Δ(Δx(t-1)), L(t) is an n×n real symmetric matrix for each t∈Z, and V(t,·)∈C1(Rn, R) for each t∈Z with V’x(t,0)=0. Our results can be regarded as a discrete analog of Ding’s results obtained in [27]. In the previous work about the existence of homoclinic orbits of difference equations, the assumption that L(t) is positive definite for each t∈Z was required. In the present paper, we try to weaken this assumption by the spectral theory of difference operators. Assume that V(t,·) is superquadratic or subquadratic. We do not assume that L and V are periodic functions, and L(t) is positive definite for all t∈Z. We show that there exists at least one non-trivial homoclinic orbit of the difference equation. Further, if V(t,x) is superquadratic and even with respect to x, then it has infinitely many different non-trivial homoclinic orbits. At the end of this chapter, two illustrative examples are provided.In Chapter3, we investigate the existence of multibump solutions of the second-order discrete Hamiltonian systems in the form of (*). We assume that V(t,·) is a sign-changing function, L and V are periodic functions. First, we show the existence of homoclinic orbit. Then, we discuss the multibump solu-tions with the assumption on the isolation of critical points by the variational methods. The study of the multibump solutions of the difference equations is just starting. As we know, there are very few results on the research of multibump solutions of homoclinic orbits of difference equations. Our results can be regarded as a discrete analog of Caldiroli and Montecchiari’s results obtained in [14].In Chapter4, we study the existence of the heteroclinic orbits of the following type of second-order discrete Hamiltonian systems: Δ2x(t-1)-μL(t)x(t)+W’x(t, x(t),δ)=0, t∈Z, where W(t,x,δ)=a(t)V(x,δ), x∈Rn, L(t) is a positive definite matrix for any t∈Z, a(·):Z�'R is a periodic function, V(·,δ)∈C2(Rn,R), and V(x,·) is continuous, and μ∈[0,1] and δ∈[0, δ0] are parameters with δ0>0. Under certain conditions, we show that for sufficiently small δ and μ, there exists at least one heteroclinic orbit connecting two critical points of the function V(x,δ). When μ=0, our results can be regarded as a discrete analog of Coti-Zelati and Rabinowitz’s results obtained in [21].As we know, the previous works on the existence of heteroclinic orbits of difference equations are about the case that μ=0and the heteroclinic orbits connecting two critical points of V with the same value, that is, the two critical points are at the same energy level [88,79,80]. For the work on the existence of heteroclinic orbits of continuous systems, there is no result about the case μ≠0.