The Kramers Oscillator Revisited

on the occasion of his 50th birthday Abstract In their 1993 paper 16], Schimansky-Geier and Herzel discovered numerically that the Kramers oscillator (which is identical with the Duung oscillator forced by additive white noise) has a positive top Lyapunov exponent in the low damping regime. In this paper, we study the Kramers oscillator from the point of view of random dynamical systems, to which we give a brief introduction. In particular, we connrm the ndings in the paper 16] about the Lyapunov exponent by performing more precise simulations, revealing that the Lya-punov exponent is positive up to a critical value of the damping, from which on it remains negative. We then show that the Kramers oscillator has a global random at-tractor which in the stable regime (large damping) is just a random point and in the unstable regime (small damping) has very complicated geometrical structure. In the latter case the invariant measure supported by the attractor is a Sinai-Ruelle-Bowen measure with positive entropy. The Kramers oscillator hence undergoes a stochastic bifurcation at the critical value of the damping parameter.