Limit cycles appearing from the perturbation of a cubic isochronous center
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917953361936384 |
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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 |