Nonintegrability and Chaos in Hamiltonian Systems with Saddle-Centers (Symmetry and Singularity of Geometric Structures and Differential Equations)

Abstract

In general, a Hamiltonian system is nonintegrabe if chaotic dynamics occurs. However, chaotic dynamics may not occur even if it is nonintegrable. Here we are interested in the following question: Does chaotic dynamics occur in a Hamiltonian system when it is nonintegrable? We review some previous results related to this question for two-degree-of-freedom Hamiltoniansystems with saddle-centers and homoclinic orbits. We also state some extensions of the results to a higher-order approximation, heteroclinic orbits and more-or infinite-degree-of-freedom systems. In particular, the extended theory shows that Arnold diffusion type motions can occur in three-or more-degree-of-freedom systems.