Existence and Multiplicity Results for Homoclinic Orbits of Hamiltonian Systems

Homoclinic orbits play an important role in the study of qualitative behavior of dynamical systems. Such kinds of orbits have been studied since the time of Poincar'e. In this paper, we discuss how to use variational methods to study the existence of homoclinic orbits of Hamiltonian systems. Introduction In the theory of differential equations, a trajectory which is asymptotic to a constant state as jtj !1 is called a homoclinic orbit. Such kinds of orbits have been found in various models of dynamical systems and they frequently have tremendous effect to the dynamics of such nonlinear systems. The homoclinic orbits have been studied since the time of Poincar'e but mainly by perturbation methods. It is only relatively recently that some new tools have been developed in the calculus of variations to show the existence of homoclinic solutions of nonlinear differential equations. In this article, a class of second order Hamiltonian systems is considered: q \\Gamma L(t)q + V 0 (t; q) = ...