Limit cycles appearing from the perturbation of a cubic isochronous center

TL;DR

This paper addresses the problem of limit-cycle bifurcation from the cubic isochronous center $S^*_2$ under $n$th-degree polynomial perturbations. It leverages the known first integral $H(x,y)=(x^2+y^2)(1+2xy)^{-1}$ and its integrating factor to form the Abelian integral $I(h)$ on the symmetric period annulus, transforming the perturbation problem into zero counting. The authors show that $I(h)$ has the structure $I(h)=h^2 I_1(h)$ and, by applying a lemma on trigonometric integrals and relating components to $L_k(h)$, obtain an upper bound of $4\left[\frac{n+1}{2}\right]+1$ zeros, hence at most that many limit cycles. This result extends the understanding of isochronous center perturbations and complements known results for $S^*_1$, while leaving open the cases for $S^*_3$ and $S^*_4$ due to the complexity of their first integrals.

Abstract

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.