Bifurcation of limit cycles from a cubic reversible isochrone

For the polynomial differential system ˙ x = −y + Chavarriga and García proved that when certain parameters of the system satisfy specific conditions, the origin is an isochronous center if and only if the system can be transformed into one of the five types: CR 1 , CR 2 , CR 3 , CR 4 , or CR 5. However, the bifurcation of limit cycles in these five isochronous systems has remained unexplored. In this paper, we focus on studying the limit cycle bifurcation of the CR 5 system under polynomial Liénard-type perturbations. Using Abelian integrals, we derive an upper bound for the number of limit cycles that can emerge from such perturbations and verify the existence of limit cycles for n = 1, 2, 3 through numerical simulations. The method we employed to obtain the algebraic structure of the Abelian integral differs in many aspects from other approaches. MSC 34C07; 34C05