Simultaneous bifurcation of limit cycles for Piecewise Holomorphic systems

TL;DR

This paper investigates how limit cycles bifurcate simultaneously in piecewise holomorphic systems, providing integral formulas and demonstrating the existence of non-nested limit cycles under perturbations.

Contribution

It introduces an Abelian integral-like expression for bifurcation analysis and proves the first existence of non-nested limit cycles in such systems.

Findings

Integral expression controls bifurcating limit cycles.

Zeros of the integral determine bifurcation points.

First proof of non-nested limit cycles in piecewise holomorphic systems.

Abstract

Let $\dot{z}=f(z)$ be a holomorphic differential equation with center at $p$. In this paper we are concerned about studying the piecewise perturbation systems $\dot{z}=f(z)+\epsilon R^\pm(z,\overline{z}),$ where $R^\pm(z,\overline{z})$ are complex polynomials defined for $\pm\operatorname{Im}(z)> 0.$ We provide an integral expression, similar to an Abelian integral, for the period annulus of $p.$ The zeros of this integral control the bifurcating limit cycles from the periodic orbits of this annular region. This expression is given in terms of the conformal conjugation between $\dot{z}=f(z)$ and its linearization $\dot{z}=f'(p)z$ at $p$. We use this result to control the simultaneous bifurcation of limit cycles of the two annular periods of $\dot{z}={\rm i} (z^2-1)/2$, after both complex and holomorphic piecewise polynomial perturbations. In particular, as far as we know, we provide the first proof of the existence of non nested limit cycles for piecewise holomorphic systems.