Stationary probability distribution near stable limit cycles far from Hopf bifurcation points

We obtain analytic results for the stationary probability distribution in the vicinity of a stable limit cycle for Markov systems described by a Fokker-Planck equation or a birth-death master equation. The results apply best for ranges of parameters removed from Hopf bifurcation points. As a by-product, we demonstrate that there holds a Liouville-like theorem for the stationary probability distribution: the product of the velocity along the limit cycle times the area of the cross section of the probability distribution transverse to the cycle is a constant. A numerical simulation of a chemical model system with a limit cycle shows good agreement with the analytic results.