LYAPUNOV EXPONENTS AND STABILITY FOR THE STOCHASTIC DUFFING-VAN DER POL OSCILLATOR

摘要

Let λ denote the almost sure Lyapunov exponent obtained by linearizing the stochastic Duffing-van der Pol oscillator xe = -ω~2x + βx - Ax~3 - Bx~2x + σxW_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 R~2 {(0, 0)} with stationary probability measure μ, say. For λ > 0 let λ 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 λ 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 λ in the presence of small noise and small viscous damping.