Slow divergence integral in regularized piecewise smooth systems

TL;DR

This paper introduces the concept of slow divergence integral along sliding segments in regularized piecewise smooth systems, demonstrating its invariance and applying it to analyze limit cycles in a specific two-fold model.

Contribution

It generalizes the slow divergence integral to regularized systems with tangency points and shows its invariance under smooth transformations, with applications to limit cycle analysis.

Findings

Invariant slow divergence integral along sliding segments.

Connection between Minkowski dimension and sliding limit cycles.

Application to a model with visible-invisible two-folds.

Abstract

In this paper we define the notion of slow divergence integral along sliding segments in regularized planar piecewise smooth systems. The boundary of such segments may contain diverse tangency points. We show that the slow divergence integral is invariant under smooth equivalences. This is a natural generalization of the notion of slow divergence integral along normally hyperbolic portions of curve of singularities in smooth planar slow-fast systems. We give an interesting application of the integral in a model with visible-invisible two-fold of type $VI_3$. It is related to a connection between so-called Minkowski dimension of bounded and monotone "entry-exit" sequences and the number of sliding limit cycles produced by so-called canard cycles.