Limit cycles of a planar differential system via averaging theory

In this article, we consider the limit cycles of a class of planar polynomial differential systems of the form \[\dot{x}=-y+\varepsilon (1+\sin ^{n}\theta )xP(x,y)\] \[ \dot{y}=x+\varepsilon (1+\cos ^{m}\theta )yQ(x,y), \] where \(P(x,y)\) and \(Q(x,y)\) are polynomials of degree \(n_{1}\) and \(n_{2}\) respectively and \(\varepsilon\) is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of a linear center \( \dot{x}=-y, \dot{y}=x,\) by using the averaging theory of first order.