Lyapunov exponents for small random perturbations of Hamiltonian systems

• 成果类型: Article
• 署名作者: Baxendale, PH; Goukasian, L
• 署名单位: University of Southern California
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2002
• 页码: 101-134
• 关键词: stochastic hopf-bifurcation; oscillator

摘要

Consider the stochastic nonlinear oscillator equation (x) double over dot = -x - x(3) + epsilon(2)betax + epsilonsigmax(W) over dot(t) with beta < 0 and sigma not equal 0. If 4beta + sigma(2) > 0 then for small enough epsilon > 0 the system (x, (x) over dot ) is positive recurrent in R-2 \ {(0, 0)}. Now let (λ) over bar(epsilon) denote the top Lyapunov exponent for the linearization of this equation along trajectories. The main result asserts that (λ) over bar(epsilon) = epsilon(2/3)(λ) over bar+ O(epsilon(4/3)) as epsilon --> 0 with (λ) over bar > 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 snore 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.