Stochastic stabilization of uncontrolled and controlled Duffing–van der Pol systems under Gaussian white-noise excitation

In this paper, the stochastic stability of uncontrolled and controlled Duffing–van der Pol systems under Gaussian white-noise excitation is investigated. On the one hand, Lyapunov exponent as a measure is used to estimate the local stability with probability one for the trivial solution of uncontrolled and controlled systems. The difference in Lyapunov exponents between these two kinds of systems is given. On the other hand, the boundary classification of Hamiltonian as a criterion is chosen to judge the global stability of coupled Duffing–van der Pol systems. And the Hamiltonians associated with controlled and uncontrolled systems are also expressed.