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

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 e-* 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 V I3). We use this framework to show that the number of limit cycles in regularized piecewise smooth polynomial systems is unbounded.(c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Automated summary

In this paper, the authors extend the slow divergence-integral to smooth systems that approach piecewise smooth ones. The slow divergence-integral is based on a generalized canard cycle for a piecewise smooth bifurcation, and it is used to show that the number of limit cycles in regularized piecewise smooth polynomial systems is unbounded.