For the polynomial differential system \(\begin{aligned} \dot{x}=-y+\sum \limits _{i+j=2}^3a_{i,j}x^iy^j,\ \dot{y}=x+\sum \limits _{i+j=2}^3b_{i,j}x^iy^j,\ a_{i,j},b_{i,j}\in \mathbb {R}, \end{aligned}\) Chavarriga and García proved that, under specific parameter conditions, the origin is an isochronous center if and only if the system can be transformed into one of five types: \(CR_1, CR_2, CR_3, CR_4\) , or \(CR_5\) . However, the bifurcation of limit cycles in these five isochronous differential systems remains unexplored. In this paper, we investigate the limit cycle bifurcation of the \(CR_5\) system under nth-degree polynomial Liénard-type perturbations. Using the Abelian integral, 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.
