Bifurcations of grazing loops of arbitrary tangent multiplicity in piecewise-smooth systems

TL;DR

This paper investigates bifurcations of grazing loops in piecewise-smooth systems with arbitrary tangent multiplicity, introducing new methods to analyze high-multiplicity cases where traditional Poincare maps fail.

Contribution

It develops a novel functional perturbation approach and localization method to analyze grazing loop bifurcations for high tangent multiplicity, extending previous low-multiplicity results.

Findings

Established a quantitative relationship between tangent multiplicity and bifurcation structures.

Constructed a functional perturbation framework for high-multiplicity grazing bifurcations.

Proposed a localization method to analyze recurrences when Poincare maps are not applicable.

Abstract

In piecewise-smooth differential systems, a hyperbolic limit cycle of a subsystem loses its structural stability if it grazes the switching manifold at a tangent point. Such a cycle is called a grazing loop and in this paper we investigate its bifurcations for arbitrary tangent multiplicity. For the low-multiplicity tangency, the recurrences are comprehensively captured by a functional perturbation with two parameters in previous publications, where the parameters characterize the recurrences near the tangent point and the limit cycle respectively. However, for high-multiplicity tangency, these parameters fail to capture the recurrences and thus, Poincare return maps can not be defined as usual. To address these challenges, we construct a functional perturbation with functions to clarify the recurrences and simultaneously, propose a localization method to make these two recurrences equivalent. We finally establish a quantitative relationship between the multiplicity of tangency and the numbers of crossing limit cycles, sliding loops bifurcating from the grazing loop and the number of tangent points on these sliding loops.