Averaging theory and catastrophes

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.