Limit cycles appearing from the perturbation of a cubic isochronous center

Fuente: arXiv
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Autores principales: Yang, Jihua, Zhang, Qipeng
Formato: Preprint
Publicado: 2025
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author Yang, Jihua
Zhang, Qipeng
author_facet Yang, Jihua
Zhang, Qipeng
contents For a polynomial differential system $$\dot{x}=-y+\sum\limits_{i+j=3}α_{i,j}x^iy^j,\quad \dot{y}=x+\sum\limits_{i+j=3}β_{i,j}x^iy^j,$$ Pleshkan (Differ. Equations, 1969) proved that the origin is an isochronous center of this system iff it can be brought to one of $S^*_1$, $S^*_2$, $S^*_3$ or $S^*_4$. The bifurcation of limit cycles for these four types of isochronous differential systems have not yet been studied, except for $S^*_1$. This paper is devoted to study the limit cycle problem of $S^*_2$ when we perturb it with an arbitrary polynomial vector field. An upper bound of the number of limit cycles is obtained using the Abelian integral.
format Preprint
id arxiv_https___arxiv_org_abs_2503_09014
institution arXiv
publishDate 2025
record_format arxiv
spellingShingle Limit cycles appearing from the perturbation of a cubic isochronous center
Yang, Jihua
Zhang, Qipeng
Dynamical Systems
For a polynomial differential system $$\dot{x}=-y+\sum\limits_{i+j=3}α_{i,j}x^iy^j,\quad \dot{y}=x+\sum\limits_{i+j=3}β_{i,j}x^iy^j,$$ Pleshkan (Differ. Equations, 1969) proved that the origin is an isochronous center of this system iff it can be brought to one of $S^*_1$, $S^*_2$, $S^*_3$ or $S^*_4$. The bifurcation of limit cycles for these four types of isochronous differential systems have not yet been studied, except for $S^*_1$. This paper is devoted to study the limit cycle problem of $S^*_2$ when we perturb it with an arbitrary polynomial vector field. An upper bound of the number of limit cycles is obtained using the Abelian integral.
title Limit cycles appearing from the perturbation of a cubic isochronous center
topic Dynamical Systems
url https://arxiv.org/abs/2503.09014