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

The Lyapunov exponents of the linearization  x = x + _ x + t x of a noisy Dung-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 1 an associated diusion with drift given by a potential function and additive noise, and, consequently, in terms of hypergeometric functions. This representation leads to dierent kinds of complete and explicit asymptotic expansions, as well as a rather complete account of global properties of the Lyapunov exponents as functions of and . Key words and phrases: Lyapunov exponents; noisy damped linear oscillator; Dung-van der Pol oscillator; Furstenberg-Khasminskii formula; invariant measure; diusion on the circle; asymptotic expansion. 1991 AMS subject classications: primary 60 H 10, 34 D 08; secondary: 60 J 60, 35 B 32. Introduction ...