Averaging theory and catastrophes

TL;DR

This paper explores how bifurcations in averaged guiding systems influence the original non-autonomous systems, revealing that certain universal bifurcations persist while others do not, with implications for understanding complex dynamical behaviors.

Contribution

It extends averaging theory by analyzing the persistence of non-hyperbolic bifurcations in non-autonomous systems, especially focusing on $\\mathcal{K}$-universal bifurcations.

Findings

$\\mathcal{K}$-universal bifurcations persist in the original system

Non-versal bifurcations like transcritical and pitchfork do not persist

Illustrated with examples including fold, transcritical, pitchfork, and saddle-focus bifurcations

Abstract

When a dynamical system is subject to a periodic perturbation, the averaging method can be applied to obtain an autonomous leading order "guiding system", placing the time dependence at higher orders. Recent research focused on investigating invariant structures in non-autonomous differential systems arising from hyperbolic structures in the guiding system, such as periodic orbits and invariant tori. Complementarily, the effect that bifurcations in the guiding system have on the original non-autonomous one has also been recently explored, albeit less frequently. This paper extends this study by providing a broader description of the dynamics that can emerge from non-hyperbolic structures of the guiding system. Specifically, we prove here that $\mathcal{K}$-universal bifurcations in the guiding system `persist' in the original non-autonomous one, while non-versal bifurcations, such as the transcritical and pitchfork, do not. We illustrate the results on examples of a fold, a transcritical, a pitchfork, and a saddle-focus.