Geometric phase for nonlinear oscillators from perturbative renormalization group

TL;DR

This paper develops a renormalization group method to analyze geometric phases in nonlinear oscillators, applicable to both autonomous and nonautonomous systems, demonstrated on Van der Pol and Duffing models.

Contribution

It introduces a novel RG-based framework to compute geometric phases in nonlinear oscillators with time-dependent parameters.

Findings

RG equations determine geometric phase during parameter variation

Method applies to Van der Pol and Van der Pol-Duffing models

Provides a unified approach for autonomous and nonautonomous oscillators

Abstract

We formulate a renormalization group approach to a general nonlinear oscillator problem. The approach is based on the exact group law obeyed by solutions of the corresponding ordinary differential equation. We consider both the autonomous models with time-independent parameters, as well as nonautonomous models with slowly varying parameters. We show that the renormalization group equations for the nonautonomous case can be used to determine the geometric phase acquired by the oscillator during the change of its parameters. We illustrate the obtained results by applying them to the Van der Pol, and Van der Pol-Duffing models.