Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 31A / b / Solution

For the unperturbed oscillator takeand energy $E=(x^2+y^2)/2$. Its change over one nearly circular orbit isA nonzero balance therefore requireswhich exists exactly when $a<1/3$. The averaged energy drift is positive below this radius and negative above it, so the resulting limit cycle is stable.