Lyapunov exponents for small random perturbations of Hamiltonian systems

Abstract

Consider the stochastic nonlinear oscillator equation ẍ = -x - x3 + ε2β ẋ + ε σ x Ẇt with β < 0 and σ ≠ 0. If 4β + σ2 > 0 then for small enough ε > 0 the system (x, ẋ) is positive recurrent in R2\{(0,0}. Now let λ̄(ε) denote the top Lyapunov exponent for the linearization of this equation along trajectories. The main result asserts that λ̄(ε) = ε2/3λ̄ + O(ε4/3) as ε → 0 with λ̄ > 0. This result depends crucially on the fact that the system above is a small perturbation of a Hamiltonian system. The method of proof can be applied to a more general class of small perturbations of two-dimensional Hamiltonian systems. The techniques used include (i) an extension of results of Pinsky and Wihstutz for perturbations of nllpotent linear systems, and (ii) a stochastic averaging argument involving motions on three different time scales.