Canard cycles of non-linearly regularized piecewise smooth vector fields

TL;DR

This paper investigates limit cycles, including canard cycles, in non-linear regularizations of planar piecewise smooth systems with fold points, providing criteria for their existence and bounds, and constructing examples with multiple limit cycles.

Contribution

It introduces a simple criterion for canard limit cycles in regularized systems and constructs examples with arbitrarily many limit cycles, advancing understanding of bifurcations in nonsmooth systems.

Findings

Criterion for existence of canard limit cycles based on zeros of the slow divergence integral

Construction of regularized systems with arbitrarily many limit cycles

Analysis of saddle-node bifurcations of canard cycles

Abstract

The main purpose of this paper is to study limit cycles in non-linear regularizations of planar piecewise smooth systems with fold points (or more degenerate tangency points) and crossing regions. We deal with a slow fast Hopf point after non-linear regularization and blow-up. We give a simple criterion for upper bounds and the existence of limit cycles of canard type, expressed in terms of zeros of the slow divergence integral. Using the criterion we can construct a quadratic regularization of piecewise linear center such that for any integer $k>0$ it has at least $k+1$ limit cycles, for a suitably chosen monotonic transition function $\varphi_k:\mathbb{R}\rightarrow\mathbb{R}$. We prove a similar result for regularized invisible-invisible fold-fold singularities of type II$_2$. Canard cycles of dodging layer are also considered, and we prove that such limit cycles undergo a saddle-node bifurcation.