Dynamics of nonlinear resonances in Hamiltonian systems

TL;DR

This paper investigates the role of nonlinear resonances in Hamiltonian systems, revealing that most resonance clusters are integrable and proposing an improved method for analyzing these systems.

Contribution

It introduces a novel approach to construct resonance clusters and demonstrates their integrability, enhancing analysis techniques for Hamiltonian systems with cubic Hamiltonians.

Findings

Most resonance clusters are integrable systems.

Construction of clusters improves analysis efficiency.

Method outperforms Galerkin method for these systems.

Abstract

It is well known that the dynamics of a Hamiltonian system depends crucially on whether or not it possesses nonlinear resonances. In the generic case, the set of nonlinear resonances consists of independent clusters of resonantly interacting modes, described by a few low-dimensional dynamical systems. We show that 1) most frequently met clusters are described by integrable dynamical systems, and 2) construction of clusters can be used as the base for the Clipping method, substantially more effective for these systems than the Galerkin method. The results can be used directly for systems with cubic Hamiltonian.