Limit cycles appearing from the perturbation of a cubic isochronous center

Abstract

For a polynomial differential system $$\dot{x}=-y+\sum\limits_{i+j=3}\alpha_{i,j}x^iy^j,\quad \dot{y}=x+\sum\limits_{i+j=3}\beta_{i,j}x^iy^j,$$ 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.