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

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.

Solved by gpt-5.6-sol high.