Emergence of Multiple Crossing Limit Cycles in Planar Piecewise Systems with Isochronous Centers and Nonsmooth Switching Manifolds

TL;DR

This paper investigates how nonsmooth switching manifolds in planar piecewise systems lead to complex dynamics, including multiple crossing limit cycles, by deriving bounds and explicit examples.

Contribution

It extends previous results to nonregular curves, providing explicit bounds and examples of multiple crossing limit cycles in nonsmooth systems.

Findings

Derived upper bounds for crossing limit cycles in specific systems.

Constructed explicit examples with four crossing limit cycles.

Showed increased complexity due to nonsmooth switching manifolds.

Abstract

Discontinuous piecewise differential systems exhibit dynamical behaviors with no counterpart in smooth systems, particularly in the presence of nonsmooth switching structures. In this work, we extend previous results for systems separated by a straight line to the case where the switching manifold is a nonregular curve, showing that the loss of regularity significantly increases the algebraic complexity of the closing conditions defining crossing limit cycles. As a consequence, we derive explicit upper bounds for the number of crossing limit cycles in planar systems formed by a linear Hamiltonian saddle and quadratic isochronous centers, and construct explicit examples exhibiting four crossing limit cycles in each case, thereby providing sharp constructive lower bounds. While the upper bounds follow from classical algebraic arguments, the realization of multiple crossing limit cycles requires solving nonlinear systems of high degree and remains highly nontrivial. These results highlight how nonsmooth switching manifolds enhance dynamical complexity and promote multistability in discontinuous piecewise systems