Limit cycles appearing from the perturbation of a cubic isochronous center

Abstract

For a polynomial differential system $$\dot{x}=-y+\sum _{i+j=3}{\alpha }_{i,j}{x}^i{y}^j,\dot{y}=x+\sum _{i+j=3}{\beta }_{i,j}{x}^i{y}^j,$$ Pleshkan (Differ. Equations, 1969) proved that the origin is an isochronous center of this system iff it can be brought to one of $S_1^{\ast }$, $S_2^{\ast }$, $S_3^{\ast }$ or $S_4^{\ast }$. The bifurcation of limit cycles for these four types of isochronous differential systems have not yet been studied, except for $S_1^{\ast }$. This paper is devoted to study the limit cycle problem of $S_2^{\ast }$ 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.