Sliding Bifurcation in Planar Piecewise Smooth Systems with Two Zones Separated by an Ellipse

Abstract

The objective of this paper is to study the sliding bifurcation in a planar piecewise smooth system with an elliptic switching curve. Some new phenomena are observed, such as a crossing limit cycle containing four intersections with the switching curve, sliding cycles having four sliding segments, and sliding cycles consisting of the entire switching curve. Firstly, we investigate the bifurcation of sliding cycle from a sliding heteroclinic connection to two cusps and show the appearance of one sliding cycle with two folds. To plot the bifurcation diagram, a planar piecewise linear system with two zones separated by an ellipse are considered. Moreover, we study in more detail the unfolding of a sliding cycle connecting four cusps by exhibiting its complete bifurcation diagram. More precisely, we explore the necessary and sufficient conditions for the existence of limit cycles and derive the concrete bifurcation curves. Additionally, a simple piecewise smooth system with nonlinear subsystems is studied, which shows the possibility of the existence of two nested limit cycles. Finally, numerical simulations are given to confirm the theoretical analysis.