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Simultaneous bifurcation of limit cycles for Piecewise Holomorphic systems

Armengol Gasull, Gabriel Rondón, Paulo R. da Silva

TL;DR

The paper addresses the problem of bifurcating limit cycles in piecewise perturbations of a holomorphic vector field with a center, introducing an Abelian-integral–style expression for the period annulus via a conformal linearization. By applying averaging theory, it derives first-order averaged functions $M_1$ and $N_1$ and analyzes their zeros to predict simultaneous bifurcations in two annuli, with explicit results for $f(z)=\mathrm{i}(z^2-1)/2$ and perturbations up to degree $m\le 3$, including both holomorphic and polynomial cases. The authors establish upper bounds on the number of zeros using the theory of extended Chebyshev systems, and demonstrate the first examples of two nests of limit cycles in PWHS, thus extending known smooth and planar results to piecewise holomorphic dynamics. The work provides a robust framework for designing and counting simultaneous bifurcations by controlling the zeros of $M_1$ and $N_1$, with potential applications to complex-analytic perturbations and discontinuous dynamical 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 $\dot{z}=f(z)+εR^\pm(z,\overline{z}),$ 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.

Simultaneous bifurcation of limit cycles for Piecewise Holomorphic systems

TL;DR

The paper addresses the problem of bifurcating limit cycles in piecewise perturbations of a holomorphic vector field with a center, introducing an Abelian-integral–style expression for the period annulus via a conformal linearization. By applying averaging theory, it derives first-order averaged functions and and analyzes their zeros to predict simultaneous bifurcations in two annuli, with explicit results for and perturbations up to degree , including both holomorphic and polynomial cases. The authors establish upper bounds on the number of zeros using the theory of extended Chebyshev systems, and demonstrate the first examples of two nests of limit cycles in PWHS, thus extending known smooth and planar results to piecewise holomorphic dynamics. The work provides a robust framework for designing and counting simultaneous bifurcations by controlling the zeros of and , with potential applications to complex-analytic perturbations and discontinuous dynamical systems.

Abstract

Let be a holomorphic differential equation with center at . In this paper we are concerned about studying the piecewise perturbation systems where are complex polynomials defined for We provide an integral expression, similar to an Abelian integral, for the period annulus of 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 and its linearization at . We use this result to control the simultaneous bifurcation of limit cycles of the two annular periods of , 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.
Paper Structure (10 sections, 14 theorems, 62 equations, 1 figure)

This paper contains 10 sections, 14 theorems, 62 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and proof of Theorems \ref{['tma']}
  3. The averaging method
  4. A miscellany of results
  5. Proof of Theorem \ref{['tma']}
  6. The bifurcations functions $M_1$ and $N_1$
  7. Proof of Theorem \ref{['tmb']} and the polynomial case
  8. Proof of Theorem \ref{['tmc']}
  9. Appendix
  10. Acknowledgements

Key Result

Theorem A

Consider the piecewise complex system eq_pert. Suppose that $\phi$ is the linearizing change of $\dot{z}=f(z)$ at $p$ such that $\phi(\Sigma)\subset\Sigma$. Then, its first order Melnikov function is $M_1(r)=M_1^+(r)-M_1^-(r),$ where In particular, each simple zero $r=r_0$ of $M_1$ provides, for $\epsilon$ sufficiently small, a hyperbolic limit cycle of eq_pert that tends to $r=r_0$ when $\epsilo

Figures (1)

  • Figure 1: Conformal map $\phi(z)=\frac{1+z}{1-z}$.

Theorems & Definitions (23)

  • Theorem A
  • Theorem B
  • Theorem C
  • Corollary 1
  • Theorem 2
  • Lemma 3
  • Theorem 4
  • Lemma 5
  • Theorem 6: Descartes Theorem
  • Proposition 7
  • ...and 13 more