Spectral Properties of Trajectories in Near Integrable Hamiltonian Systems

Abstract

Spectral properties have often been used in the study of trajectories of dynamical systems. It is well known that the spectrum of a quasiperiodic motion is characterized by a few distinct major peaks, located at positions which are integer combinations of the fundamental frequencies, instead of a continuous distribution of peaks which appears in the case of chaotic motion [1–4], in which case the appearance of 1/f noise has been shown in Hamiltonian systems of two degrees of freedom [5,6]. Spectral properties of trajectories associated with the deformation of tori in near integrable Hamiltonian systems are studied in [7]. The purpose of this paper is to apply this method on a system of three degrees of freedom in order to study the transition from regular motion to chaos. The method is based on the spectral analysis of the integrals of motion of an integrable Hamiltonian along trajectories of the non-integrable perturbed system.