Lyapunov Exponents and Stability for the Stochastic Duffing-Van der Pol Oscillator

Abstract

Let λ denote the almost sure Lyapunov exponent obtained by linearizing the stochastic Duffing-van der Pol oscillator

$$ \ddot{x} = - {w^{2}}x + \beta \dot{x} - A{x^{3}} - B{x^{2}}\dot{x} + \sigma x{\dot{W}_{t}} "$$

at the origin x = x = 0 in phase space. If λ > 0 then the process {(x _t , x _t ): t > 0} is positive recurrent on R² {(0,0)} with stationary probability measure μ, say. For λ > 0 let $\tilde{\lambda }$ denote the almost sure Lyapunov exponent obtained by linearizing the same equation along a typical stationary trajectory in R ² ∖{(0,0)}. The sign of $\tilde{\lambda }$ is important for stability properties of the two point motion associated with the original equation. We use stochastic averaging techniques to estimate the value of $\tilde{\lambda }$ in the presence of small noise and small viscous damping.