Nonpersistence of periodic orbits, homoclinic orbits, first integrals and commutative vector fields in perturbed systems (Symmetry and Singularity of Geometric Structures and Differential Equations)

Abstract

Determination of whether periodic orbits, homoclinic orbits, first integrals or commutative vector fields may persist under perturbations is one of the most important problems in the field of dynamical systems. In this paper, we give several theorems on necessary conditions for their persistence in general perturbed systems. Moreover, we consider periodic perturbations of one-degree-of-freedom Hamiltonian systems and describe some relationships between our results and the standard Melnikov method for periodic orbits and homoclinic orbits. This is a joint work with Kazuyuki Yagasaki (Kyoto University).