Bifurcation analysis of multiple limit cycles created in boundary equilibrium bifurcations in hybrid systems

TL;DR

This paper analyzes how multiple limit cycles emerge in hybrid systems during boundary equilibrium bifurcations, providing a general unfolding theory and illustrating it with high-dimensional examples.

Contribution

It introduces a comprehensive bifurcation analysis of multiple limit cycles created at boundary equilibrium bifurcations in hybrid systems of any dimension.

Findings

Two codimension-two bifurcation scenarios are characterized.

Bifurcation diagrams feature saddle-node or period-doubling bifurcation curves.

Theoretical results are validated with high-dimensional examples.

Abstract

A boundary equilibrium bifurcation (BEB) in a hybrid dynamical system occurs when a regular equilibrium collides with a switching surface in phase space. This causes a transition to a pseudo-equilibrium embedded within the switching surface, but limit cycles (LCs) and other invariant sets can also be created and the nature of these is not well understood for systems with more than two dimensions. This work treats two codimension-two scenarios in hybrid systems of any number of dimensions, where the number of small-amplitude limit cycles bifurcating from a BEB changes. The first scenario involves a limit cycle (LC) with a Floquet multiplier $1$ and for nearby parameter values the BEB creates a pair of limit cycles. The second scenario involves a limit cycle with a Floquet multiplier $-1$ and for nearby parameter values the BEB creates a period-doubled solution. Both scenarios are unfolded in a general setting, showing that typical two-parameter bifurcation diagrams have a curve of saddle-node or period-doubling bifurcations emanating transversally from a curve of BEBs at the codimension-two point. The results are illustrated with three-dimensional examples and an eight-dimensional airfoil model. Detailed computational results show excellent agreement to the unfolding theory and reveal further interesting dynamical features that remain to be explored.