Limit cycles in piecewise smooth systems with circular switching manifold

TL;DR

This paper analyzes limit cycles in piecewise smooth systems with circular switching manifolds, establishing transformations, bounds, and explicit examples to advance understanding of their dynamics.

Contribution

It introduces a M"obius transformation linking circular and straight-line discontinuities, derives bounds on limit cycles, and constructs explicit algebraic examples.

Findings

Established an equivalence between circular and straight-line discontinuities preserving key properties.

Derived lower bounds on the number of limit cycles for polynomial holomorphic systems.

Proved upper bounds for antiholomorphic systems and constructed explicit algebraic limit cycles.

Abstract

We study limit cycles in piecewise complex systems with switching manifold $\mathbb{S}^1$. Using M\"obius transformations we establish an equivalence between circular and straight-line discontinuities that preserves periods, stability, and algebraic structure. For piecewise polynomial holomorphic systems we obtain lower bounds on the number of limit cycles via second-order averaging and, for low degrees, via Lyapunov quantities. For piecewise antiholomorphic systems we prove upper bounds: at most $3$ limit cycles in the linear case and $10$ in the quadratic case. We also prove a rigidity theorem: when both components admit classical holomorphic normal forms at the origin no crossing limit cycles exist. Finally, we construct explicit algebraic limit cycles in the circular context, providing, as far as we know the first such examples in the literature.