Phase and amplitude responses for delay equations using harmonic balance

TL;DR

This paper develops a harmonic balance-based framework to construct phase and amplitude response functions for delay-differential equations, aiding the analysis of delay-induced oscillations under external forcing.

Contribution

It introduces a novel method combining harmonic balance with Floquet theory to analyze DDEs' phase and amplitude responses.

Findings

Framework successfully constructs response functions for DDEs.

Enables better understanding of delay-induced oscillations.

Facilitates analysis of external forcing effects on DDEs.

Abstract

Robust delay induced oscillations, common in nature, are often modeled by delay-differential equations (DDEs). Motivated by the success of phase-amplitude reductions for ordinary differential equations with limit cycle oscillations, there is now a growing interest in the development of analogous approaches for DDEs to understand their response to external forcing. When combined with Floquet theory, the fundamental quantities for this reduction are phase and amplitude response functions. Here, we develop a framework for their construction that utilises the method of harmonic balance.