We study the maximum number of limit cycles which bifurcate from the periodic orbits of the linear centre x˙=y \dot x = y , y˙=x \dot y = - x , when it is perturbed in the form x˙=yε(1+coslθ)P(x,y),y˙=xε(1+cosmθ)Q(x,y), \dot x = y - \varepsilon (1 + \mathop {\cos }\nolimits^l \theta )P(x,{\kern 1pt} y),\quad \dot y = - x - \varepsilon \left( {1 + \mathop {\cos }\nolimits^m \theta } \right)Q(x,{\kern 1pt} y), where ɛ > 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.

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Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics