Canard cycles of non-linearly regularized piecewise smooth vector fields

Abstract

The main purpose of this paper is to study limit cycles of non-linear regularizations of planar piecewise smooth systems. We deal with a slow-fast Hopf point after non-linear regularization and blow-up. We give a simple criterion for the existence of limit cycles of canard type blue for a class of (non-linearly) regularized piecewise smooth systems, expressed in terms of zeros of the slow divergence integral. Using the criterion we can construct a quadratic regularization of a 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 φk : R → R. We prove a similar result for regularized codimension 1 invisible-invisible fold-fold singularities of type II2. Canard cycles of dodging layer are also considered, and we prove that there can be at most 2 limit cycles (born in a saddle-node bifurcation).