A stochastic minimax optimal control strategy for uncertain quasi-Hamiltonian systems is proposed based on the stochastic averaging method, stochastic maximum principle and stochastic differential game theory. First, the partially completed averaged Itô stochastic differential equations are derived from a given system by using the stochastic averaging method for quasi-Hamiltonian systems with u...

Consider the stochastic Duffing-van der Pol equation ẍ = −ω2x− Ax −Bx2ẋ + εβẋ + εσxẆt with A ≥ 0 and B > 0. If β/2 + σ/8ω > 0 then for small enough ε > 0 the system (x, ẋ) is positive recurrent in R \ {0}. Let λ̃ε denote the top Lyapunov exponent for the linearization of this equation along trajectories. The main result asserts that λ̃ε ∼ ελ̃ as ε → 0 where λ̃ is the top Lyapunov exponent along tra...