Show that the zero order approximation to the limit cycle

Question

A limit cycle by perturbation theory Use perturbation theory to investigate the limit cycle of Rayleigh's equation, taken here in the form

x^·· + ∈((1/3)x^·2 - 1)x^· + x = 0,

where ∈ is a small positive parameter. Show that the zero order approximation to the limit cycle is a circle and determine its centre and radius. Find the frequency of the limit cycle correct to order ∈², and find the function x(t) correct to order.