Abstract
We study the maximum number of limit cycles which bifurcate from the periodic orbits of the linear centre <(x)over dot> = y; <(y)over dot> = -x, when it is perturbed in the form
<(x)over dot> = y - epsilon(1 + cos(l) theta) P(x, y), <(y)over dot> = -x - epsilon (1 + cos(m) theta) Q(x, y), (1)
where epsilon > 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 theta = 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.