Lyapunov Exponents for Small Random Perturbations of Hamiltonian Systems

Abstract

Consider the stochastic nonlinear oscillator equationwith β < 0 and σ = 0. If 4β + σ 2 > 0 then for small enough ε > 0 the system (x, ẋ) is positive recurrent in R 2 \ {(0, 0}. Now let λ(ε) denote the top Lyapunov exponent for the linearization of this equation along trajectories. The main result asserts thatas ε → 0 with λ > 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.

Stability of a Stochastic Two-Dimensional Non-Hamiltonian System

“…(Stated differently, we are studying the slowly-varying adiabatic system in the action-angle variables.) Similar results are also reported by Baxendale and Goukasian [10] for the multiplicative case, where calculations were done using the coordinates suggested by Sowers [11].…”

Stability of noisy nonlinear auto-parametric systems

“…The primary concern in this paper is the analysis of the single mode solutions of nonlinear auto-parametric systems; the determination and prediction of steady-state or stationary motions and their corresponding stability. There are recent results for the top Lyapunov exponent of weakly perturbed single oscillators with single-well potentials by Arnold et al [16] and Baxendale and Goukasian [17]. Contrary to the single oscillators, there are no results pertaining to stationary measures and their stability for coupled nonlinear oscillators.…”

“…Our aim is to obtain an asymptotic expansion of the top Lyapunov exponent of the random dynamical system described by (17) and (18) by making use of the prescribed scaling. This decomposition of the linear variational equation (10) into two four-dimensional decoupled systems (17) and (18) is solely due to property (2), and these four-dimensional problems are the starting point for the rest of our analysis. (17) The random motions in (17) consist of fast dynamics along the unperturbed trajectories of the deterministic system and slow motion across these trajectories.…”

“…and f 1 ⊥ is orthogonal to f 1 , the variational Equation (17) in its general form, can be transformed after some cumbersome algebraic manipulations to…”

“…Hence, we consider the equations of motion as dissipative and noisy perturbations of a four-dimensional Hamiltonian system. A perturbation approach based on the results of Arnold et al [16,19] and Baxendale and Goukasian [17] for additive and multiplicative excitations, respectively, is used to obtain explicit expressions for all the exponents in the presence of small intensity noise and small dissipation.…”