Finding Hannay angle in dissipative oscillatory systems via conservative perturbation theory

TL;DR

This paper adapts conservative Hamiltonian perturbation theory to analyze dissipative oscillatory systems, specifically calculating the Hannay angle for the van der Pol oscillator's limit cycle, bridging a gap in nonconservative system analysis.

Contribution

It demonstrates the use of Lie transform Hamiltonian perturbation theory for dissipative systems and analytically computes the Hannay angle in a nonconservative context.

Findings

Successfully adapted Hamiltonian perturbation theory to dissipative systems.

Analytically derived the Hannay angle for the van der Pol oscillator.

Numerically confirmed the equivalence of the geometric phase and Hannay angle.

Abstract

Usage of a Hamiltonian perturbation theory for a nonconservative system is counterintuitive and in general, a technical impossibility by definition. However, the time-independent dual Hamiltonian formalism for the nonconservative systems have opened the door for using various conservative perturbation theories for investigating the dynamics of such systems. Here we demonstrate that the Lie transform Hamiltonian perturbation theory can be adapted to find the perturbative solutions and the frequency corrections for the dissipative oscillatory systems. As a further application, we use the perturbation theory to analytically calculate the Hannay angle for the van der Pol oscillator's limit cycle trajectory when its parameters-the strength of the nonlinearity and the frequency of the linear part-evolve cyclically and adiabatically. For this van der Pol oscillator, we also numerically calculate the corresponding geometric phase and establish its equivalence with the Hannay angle.