Variational Criteria for Nonintegrability and Chaos in Hamiltonian Systems

Abstract

The goal of this article is to show how existence results obtained by variational methods can be applied to prove chaotic behavior for a class of Hamiltonian systems. The standard tool is the perturbation theory, for example, the Poincaré-Melnikov-Arnold integral. This approach is restricted to the case when the system is in some sense close to integrable. If this is not so, non-perturbative methods are needed. Our method works best for natural analytic systems with two degrees of freedom. We obtain the existence of homoclinic orbits by variational methods and then prove, using analyticity, that they are isolated from purely topological reasons. Of course, it is impossible to prove transversality by variational methods. However, if the homoclinic orbit is obtained by minimizing the action functional, or by the mountain-pass theorem, then it is usually possible to prove a weaker property, that they are transversal in the topological sense. Then we can apply a modification of the well-known results on dynamics near transversal homoclinics.