Limit cycles bifurcating from the period annulus of a uniform isochronous center in a quartic polynomial differential system

Abstract

We study the number of limit cycles that bifurcate from the periodic solutions surrounding a uniform isochronous center located at the origin of the quartic polynomial differential system $$ \dot{x}=-y+xy(x^2+y^2),\quad \dot{y}=x+y^2(x^2+y^2), $$ when perturbed in the class of all quartic polynomial differential systems. Using the averaging theory of first order we show that at least 8 limit cycles bifurcate from the period annulus of the center. Recently this problem was studied by Peng and Feng [9], where the authors found 3 limit cycles.

Keywords

• Polynomial vector field
• limit cycle
• averaging method
• periodic orbit
• uniform isochronous center