Limit cycles of a generalised Mathieu differential system

Abstract

Abstract 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 (1) x ˙ = y − ε ( 1 + cos l θ ) P ( x ,   y ) ,   y ˙ = − x − ε ( 1 + cos m θ ) 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.