  We consider stochastic differential equations, obtained by adding weak
Gaussian white noise to ordinary differential equations admitting $N$
asymptotically stable periodic orbits. We construct a discrete-time,
continuous-space Markov chain, called a random Poincar\'e map, which encodes
the metastable behaviour of the system. We show that this process admits
exactly $N$ eigenvalues which are exponentially close to $1$, and provide
expressions for these eigenvalues and their left and right eigenfunctions in
terms of committor functions of neighbourhoods of periodic orbits. The
eigenvalues and eigenfunctions are well-approximated by principal eigenvalues
and quasistationary distributions of processes killed upon hitting some of
these neighbourhoods. The proofs rely on Feynman--Kac-type representation
formulas for eigenfunctions, Doob's $h$-transform, spectral theory of compact
operators, and a recently discovered detailed-balance property satisfied by
committor functions.
