Augmented Phase Reduction for Periodic Orbits Near a Homoclinic Bifurcation and for Relaxation Oscillators

TL;DR

This paper develops analytical expressions for augmented phase reduction applicable to oscillators near homoclinic bifurcations and relaxation oscillators, validating them through simulations to identify when this method outperforms standard phase reduction.

Contribution

It provides the first analytical formulas for augmented phase reduction in these specific oscillator regimes, enhancing model reduction techniques.

Findings

Analytical expressions for augmented phase reduction were derived.

Simulations confirmed the accuracy of the analytical formulas.

Conditions identified where augmented phase reduction is more effective.

Abstract

Oscillators - dynamical systems with stable periodic orbits - arise in many systems of physical, technological, and biological interest. The standard phase reduction, a model reduction technique based on isochrons, can be unsuitable for oscillators which have a small-magnitude negative nontrivial Floquet exponent. This necessitates the use of the augmented phase reduction, a recently devised two-dimensional reduction technique based on isochrons and isostables. In this article, we calculate analytical expressions for the augmented phase reduction for two dynamically different planar systems: periodic orbits born out of a homoclinic bifurcation, and relaxation oscillators. To validate our calculations, we simulate models in these dynamic regimes, and compare their numerically computed augmented phase reduction with the derived analytical expressions. These analytical and numerical calculations help us to understand conditions for which the use of augmented phase reduction over the standard phase reduction can be advantageous.