Fractal analysis of canard cycles and slow-fast Hopf points in piecewise smooth Li\'{e}nard equations

TL;DR

This paper provides a comprehensive fractal analysis of piecewise smooth Lie9nard equations, focusing on Minkowski dimensions of orbits near Hopf points and canard cycles, revealing insights into limit cycle existence and bifurcations.

Contribution

It introduces a detailed fractal analysis framework for PWS Lie9nard systems, linking Minkowski dimension to limit cycle behavior and bifurcations near critical points.

Findings

Minkowski dimension values near Hopf points and canard cycles are characterized.

Trivial Minkowski dimension indicates non-existence of certain limit cycles.

Bounded canard cycles can undergo saddle-node bifurcations affecting limit cycle count.

Abstract

The main goal of this paper is to give a complete fractal analysis of piecewise smooth (PWS) slow-fast Li\'{e}nard equations. For the analysis, we use the notion of Minkowski dimension of one-dimensional orbits generated by slow relation functions. More precisely, we find all possible values for the Minkowski dimension near PWS slow-fast Hopf points and near bounded balanced crossing canard cycles. We study fractal properties of the unbounded canard cycles using PWS classical Li\'{e}nard equations. We also show how the trivial Minkowski dimension implies the non-existence of limit cycles of crossing type close to Hopf points. This is not true for crossing limit cycles produced by bounded balanced canard cycles, i.e. we find a system undergoing a saddle-node bifurcation of crossing limit cycles and a system without limit cycles (in both cases, the Minkowski dimension is trivial). We also connect the Minkowski dimension with upper bounds for the number of limit cycles produced by bounded canard cycles.