An explicit description of the Lyapunov exponents of the noisy damped harmonic oscillator

The Lyapunov exponents of the linearization of a noisy Duffing van der Pol oscillator are key quantities in the investigation of the stochastic Hopf bifurcation of this system. Considering the white noise case we derive a simple equation exhibiting them explicitly as functions of the fourth moment of the invariant measure of an associated diffusion with drift given by a potential function and additive noise, and, consequently, in terms of hypergeometric functions. This representation leads to different kinds of complete and explicit asymptotic expansions, as well as a rather complete account of global properties of the Lyapunov exponents as functions of (beta) and (sigma).