The number of limit cycles for regularized piecewise polynomial systems is unbounded

TL;DR

This paper extends the slow divergence-integral concept to smooth systems approaching piecewise smooth systems, demonstrating that the number of limit cycles in regularized polynomial systems can be unbounded, revealing complex bifurcation behavior.

Contribution

It introduces a generalized slow divergence-integral for piecewise smooth systems and proves the unboundedness of limit cycles in regularized polynomial systems.

Findings

Number of limit cycles in regularized systems is unbounded.

Extended slow divergence-integral to piecewise smooth bifurcations.

Demonstrated complex bifurcation behavior in polynomial systems.

Abstract

In this paper, we extend the slow divergence-integral from slow-fast systems, due to De Maesschalck, Dumortier and Roussarie, to smooth systems that limit onto piecewise smooth ones as $\epsilon\rightarrow 0$. In slow-fast systems, the slow divergence-integral is an integral of the divergence along a canard cycle with respect to the slow time and it has proven very useful in obtaining good lower and upper bounds of limit cycles in planar polynomial systems. In this paper, our slow divergence-integral is based upon integration along a generalized canard cycle for a piecewise smooth two-fold bifurcation (of type visible-invisible called $VI_3$). We use this framework to show that the number of limit cycles in regularized piecewise smooth polynomial systems is unbounded.