Simultaneous bifurcation of limit cycles for Piecewise Holomorphic systems
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866909454516092928 |
|---|---|
| author | Gasull, Armengol Rondón, Gabriel da Silva, Paulo R. |
| author_facet | Gasull, Armengol Rondón, Gabriel da Silva, Paulo R. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_06674 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simultaneous bifurcation of limit cycles for Piecewise Holomorphic systems Gasull, Armengol Rondón, Gabriel da Silva, Paulo R. Dynamical Systems Classical Analysis and ODEs Complex Variables 32A10, 34A36, 34C07, 37G15 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. |
| title | Simultaneous bifurcation of limit cycles for Piecewise Holomorphic systems |
| topic | Dynamical Systems Classical Analysis and ODEs Complex Variables 32A10, 34A36, 34C07, 37G15 |
| url | https://arxiv.org/abs/2501.06674 |