ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

12. Probability and Statistics

1991 MS Classification: 60, 34 Imkeller, Peter, Humboldt-Universität, Berlin, Germany An explicit description of the Lyapunov exponents of the noisy damped harmonic oscillator. The Lyapunov exponents of the linearization

ẍ = −x + βẋ + σξ_{t}x

of a noisy Duffing-van der Pol oscillator are key quantities in the investigation of the stochastic bifurcation behaviour of the system. We derive a simple equation exhibiting them explicitly as functions of the second 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 β and σ. Our method can be extended to more general two-dimensional systems. This is joint work with C. Lederer.