Periodic Motions In Lattice Dynamical Systems

In this thesis, our main concern is how to obtain the existence and multiplicity of nontrivial time periodic motions of finite energy for the general Fermi-Pasta-Ulam (FPU for short) type lattice dynamical systems. Our argument is variational. In our recent paper [48], under the more general assumptions on the superquadratic potentials at infinity, the well known result [7] has been recovered and improved by our method. It is worth mentioning that the most interesting result in [48] refers to the multiplicity, that is, infinitely many geometrically distinct solutions are obtained for the strongly indefinite case by extending the abstract critical point theorem about multiplicity for a strongly indefinite functional developed by Bartsch and Ding [12] to a more gener-al class of symmetry(since our functional is invariant under the action of group Z and S1, in some sense, we weaken the rather strong non-degenerate assumption to the de-generate case). More recently, in [49] we obtain the existence of ground state time periodic solutions for strongly indefinite lattice dynamical system with superquadratic potentials. In [50], we consider the existence of the time periodic motions for the both definite and strongly indefinite lattice systems with asymptotically quadratic potentials, and in [51] nontrivial periodic motions for resonant type asymptotically linear lattice dynamical systems are obtained via Morse theory. To our knowledge, there is no result dealing with these asymptotically quadratic cases.We briefly outline the organization of this dissertation. In Chapter1, we recall some the background and the latest research developments of the FPU type lattice dy-namical systems. In Chapter2, we suppose the potential Φi is superquadratic at infinity. For all T>0, we obtain a nonzero T-periodic solution of finite energy which may be nonconstant in some range of period. If in addition Φi(x) is even in x, we also obtain infinitely many geometrically distinct solutions for any period T>0. In particular, a prescribed number of geometrically distinct nonconstant periodic solutions are obtained for some range of period. In Chapter3, under the more general superquadratic assump-tions on potentials Φi, for T>0, we obtain ground state T periodic solutions, i.e., non- trivial solutions with least possible energy; we also prove the solution obtained above may be nonconstant in some range of periodic. In Chapter4, we will give a positive an-swer to the Open problem2.7in [38]). Suppose Φi(x)=-αi/2x2+Vi(x) is asymptotically quadratic at infinity, i.e., Φii(x) tends to a quadratic function as|x|â†'∞. Furthermore, we assume the coefficients at αi≠0for all i. Based on the mountain pass theorem without the compact condition and an abstract critical point theorem about existence for strong-ly indefinite functional developed by Bartsch and Ding [12], for all T>0, we obtain a nonzero T-periodic solution of finite energy. In the final Chapter5, we consider the nonautonomous lattice systems. Assume the potential Φi(t,x)=-(αi/2)|x|2+Vi(t,x) is T-periodic in t for some T>0and satisfies Φi+N=Φi for some N∈N. Moreover, sup-pose that αi=0for some i∈Z which implies this system is resonant both at origin and at infinity under some additional conditions. Based on some new results concerning the precise computations of the critical groups, for a given m∈Z, we obtain the existence of nontrivial periodic solution q satisfying qi+mN(t+T)=qt(t) for all t∈R and i∈Z.