Limit cycles of a generalised Mathieu differential system

We study the maximum number of limit cycles which bifurcate from the periodic orbits of the linear centre ̇x = y, ̇y = −x, when it is perturbed in the form x˙=y-ɛ(1+coslθ)P(x,y), y˙=-x-ɛ(1+cosmθ)Q(x,y), \dot x = y - \varepsilon \left( {1 + {{\cos }^l}\theta } \right)P\left( {x,y} \right),\,\,\,\,\dot y = - x - \varepsilon \left( {1 + {{\cos }^m}\theta } \right)Q\left( {x,y} \right), where ε > 0 is a small parameter, l and m are positive integers, P(x, y) and Q(x, y) are arbitrary polynomials of degree n, and θ = arctan(y/x). As we shall see the differential system (1) is a generalisation of the Mathieu differential equation. The tool for studying such limit cycles is the averaging theory.