Conservative perturbation theory for nonconservative systems

TL;DR

This paper extends canonical perturbation theory to dissipative systems with limit cycles, revealing Hamiltonian structures in certain nonconservative systems and opening avenues for broader application.

Contribution

It demonstrates how to apply canonical perturbation theory to nonconservative systems and identifies Hamiltonian structures within Lie9nard systems, surpassing previous limitations.

Findings

Canonical perturbation theory can be applied to dissipative systems.

Hamiltonian structures exist in certain Lie9nard systems.

Potential for extending the method to more non-conservative systems.

Abstract

In this paper, we show how to use canonical perturbation theory for dissipative dynamical systems capable of showing limit cycle oscillations. Thus, our work surmounts the hitherto perceived barrier for canonical perturbation theory that it can be applied only to a class of conservative systems, viz.,~Hamiltonian systems. In the process, we also find Hamiltonian structure for an important subset of Li\'enard system--- a paradigmatic system for modeling isolated and asymptotic oscillatory state. We discuss the possibility of extending our method to encompass even wider range of non-conservative systems.