A limit cycle by perturbation theory Use perturbation theory to investigate the limit cycle of Rayleigh's equation, taken here in the form
x¨+ϵ(31x˙2−1)x˙+x=0,
where ϵ 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 ϵ2, and find the function x(t) correct to order ϵ.