Lyapunov exponents for small random perturbations of nilpotent and Hamiltonian systems

Consider the stochastic Duffing Equation x&d3;=-x-x3 +32bx&d2; +3sxW&d2;t with b<0 and s≠0 . If 4b+s2>0 then for small enough 3>0 the system (x, x˙) is positive recurrent in R2{lcub}(0, 0{rcub}. Now let l&d1;3 denote the top Lyapunov exponent for the linearization of this equation along trajectories. The main result asserts that l&d1;3 =32/3l&d1; +O&parl0;34/3&parr0;as 3→0 with l&d1;>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 nilpotent linear systems, and (ii) a stochastic averaging argument involving motions on three different time scales.