LYAPUNOV EXPONENTS AND RESONANCE FOR SMALL PERIODIC AND RANDOM PERTURBATIONS OF A CONSERVATIVE LINEAR SYSTEM

We give algorithms for the asymptotic expansions of the almost sure and moment Lyapunov exponents associated with the two-dimensional stochastic differential equation obtained as a small perturbation of the deterministic rotation with rate ω. The matrices in the perturbation terms are all assumed to be periodic functions of time with period l. The form of the algorithms depends on whether or not the periods 2π/ω and l of the unperturbed system and the perturbation coefficients are commensurable (i.e. the ratio of the periods is rational). In the commensurable case certain resonances may cause jumps in the Lyapunov exponents. We give an example of a stochastically perturbed Hill's oscillator which is almost surely stable when 2ω is not an integer, but is almost surely unstable at resonant frequencies ω = m/2. This work extends recent results of Imkeller and Milstein.