Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 31A (Dynamical Systems)

Linearization about $X(t)$ givesOne Floquet multiplier of $\Phi (T)$ is $1$, corresponding to a phase shift along the orbit. Liouville's formula givesIn the plane this determinant is the other, transverse multiplier. The periodic orbit is transversely asymptotically stable when the displayed quantity is less than one, equivalently when the divergence integral is negative, and unstable when it is greater than one.

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.