Energy exchange and localization in essentially nonlinear oscillatory systems: canonical formalism

TL;DR

This paper introduces a canonical formalism using action-angle variables to analyze energy localization and exchange in nonlinear oscillatory systems, providing a unified and versatile framework that extends previous harmonic balance methods.

Contribution

It develops a canonical formalism for describing resonance manifolds in nonlinear oscillators, unifying and extending prior approaches like harmonic balance.

Findings

Resonance manifolds are effectively described using canonical action-angle variables.

The formalism simplifies proving conservation laws in autonomous and non-autonomous systems.

Application to coupled oscillators demonstrates the method's broad applicability.

Abstract

Over recent years, a lot of progress has been achieved in understanding of the relationship between localization and transport of energy in essentially nonlinear oscillatory systems. In this paper we are going to demonstrate that the structure of the resonance manifold can be conveniently described in terms of canonical action-angle variables. Such formalism has important theoretical advantages: all resonance manifolds may be described at the same level of complexity, appearance of additional conservation laws on these manifolds is easily proven both in autonomous and non-autonomous settings. The harmonic balance - based complexification approach, used in many previous studies on the subject, is shown to be a particular case of the canonical formalism. Moreover, application of the canonic averaging allows treatment of much broader variety of dynamical models. As an example, energy exchanges in systems of coupled trigonometrical and vibro-impact oscillators are considered.