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

The Lyapunov exponents of the linearization (x) double over dot = -x + 2 beta(x) over dot + sigma xi(t)x of a noisy Duffing-van der Pol oscillator are Key quantities in the investi gation of the stochastic Hopf bifurcation of this system. Considering the w hite noise case we derive a simple equation exhibiting them explicitly as f unctions of the fourth moment of the invariant measure of an associated dif fusion with drift given by a potential function and additive noise, and, co nsequently, 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 exponent s as functions of beta and sigma.