Unveiling the Cyclicity of Monodromic Tangential Singularities: Insights Beyond the Pseudo-Hopf Bifurcation

TL;DR

This paper explores new bifurcation phenomena in nonsmooth systems, showing that destroying certain monodromic tangential singularities produces multiple limit cycles, thus advancing understanding beyond the pseudo-Hopf bifurcation.

Contribution

It demonstrates that destroying $(2k,2k)$-monodromic tangential singularities yields at least $k$ limit cycles, extending bifurcation analysis beyond pseudo-Hopf bifurcations.

Findings

Destruction of $(2k,2k)$-monodromic tangential singularities produces at least $k$ limit cycles.

Expands understanding of bifurcations in nonsmooth systems beyond pseudo-Hopf bifurcation.

Provides theoretical insights with practical implications for systems with switches and abrupt processes.

Abstract

The cyclicity problem, crucial in analyzing planar vector fields, consists in estimating the number of limit cycles emanating from monodromic singularities. Traditionally, this estimation relies on Lyapunov coefficients. However, in nonsmooth systems, besides the limit cycles bifurcating by varying the Lyapunov coefficients, monodromic singularities on the switching curve can always be split apart yielding, under suitable conditions, a sliding region and an additional limit cycle surrounding it. This bifurcation phenomenon, known as pseudo-Hopf bifurcation, has enhanced lower-bound cyclicity estimations for monodromic singularities in Filippov systems. In this study, we push beyond the pseudo-Hopf bifurcation, demonstrating that the destruction of $(2k,2k)$-monodromic tangential singularities yields at least $k$ limit cycles surrounding sliding segments. This new bifurcation phenomenon expands our understanding of limit cycle bifurcations in nonsmooth systems and, in addition to the theoretical significance, has practical relevance in various applied models involving switches and abrupt processes.