Stochastic averaging and asymptotic behavior of the stochastic Duffing–van der Pol equation

Consider the stochastic Duffing–van der Pol equationẍ=−ω2x−Ax3−Bx2ẋ+ε2βẋ+εσxẆtwith A⩾0 and B>0. If β/2+σ2/8ω2>0 then for small enough ε>0 the system (x,ẋ) is positive recurrent in R2⧹{0}. Let λ̃ε denote the top Lyapunov exponent for the linearization of this equation along trajectories. The main result asserts thatλ̃ε∼ε2λ̃ as ε→0,where λ̃ is the top Lyapunov exponent along trajectories for a stochastic differential equation obtained from the stochastic Duffing–van der Pol equation by stochastic averaging. In the course of proving this result, we develop results on stochastic averaging for stochastic flows, and on the behavior of Lyapunov exponents and invariant measures under stochastic averaging. Using the rotational symmetry of the stochastically averaged system, we develop theoretical and numerical methods for the evaluation of λ̃. We see that the sign of λ̃, and hence the asymptotic behavior of the stochastic Duffing–van der Pol equation, depends strongly on ωB/A. This dimensionless quantity measures the relative strengths of the nonlinear dissipation Bx2ẋ and the nonlinear restoring force Ax3.