Limit cycles obtained by perturbing a degenerate center

This paper deals with the maximum number of limit cycles bifurcating from the degenerate centre \[ \dot{x}=-y(3x^2+y^2),\: \dot{y}=x(x^2-y^2), \] when we perturb it inside a class of all homogeneous polynomial differential systems of degree \(5\). Using averaging theory of second order, we show that, at most, five limit cycles are produced from the periodic orbits surrounding the degenerate centre under quintic perturbation. In addition, we provide six examples that give rise to exactly \(5, 4, 3, 2, 1\) and \(0\) limit cycles.