Lyapunov Exponents and Stability for the Stochastic Duffing-van Der Pol Oscillator

Let λ denote the almost sure Lyapunov exponent obtained by linearizing the stochastic Duffing-van der Pol oscillator ẍ = −ωx + βẋ−Ax −Bxẋ + σxẆt at the origin x = ẋ = 0 in phase space. If λ > 0 then the process {(xt, ẋt) : t ≥ 0} is positive recurrent on R \ {(0, 0)} with stationary probability measure μ, say. For λ > 0 let λ̃ denote the almost sure Lyapunov exponent obtained by linearizing the same equation along a typical stationary trajectory in R \ {(0, 0)}. The sign of λ̃ 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 λ̃ in the presence of small noise and small viscous damping.