JOURNAL OF SOUND AND VIBRATION Journal of Sound and Vibration 290 (2006) 723–735 Stochastic stabilization of uncontrolled and controlled Duffing–van der Pol systems under Gaussian white-noise excitation W. Xu à , W. Li, J.F. Zhao Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an Shaanxi 710072, PR China Received 12 November 2004; received in revised form 28 February 2005; accepted 20 April 2005 Available online 24 August 2005 Abstract 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. r 2005 Elsevier Ltd. All rights reserved. 1. Introduction In the last century, the theory of stochastic optimal control for systems under random excitations has developed rapidly with wide applications in many scientific fields, especially in economics and physics [1–4]. For the problem of stochastic stabilization control, the purpose is mainly to design a control law to make unstable random dynamic systems become stable, or to enhance the stability balance of a stable random dynamic systems. Besides, the dynamic stability ARTICLE IN PRESS www.elsevier.com/locate/jsvi 0022-460X/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2005.04.010 à Corresponding author. Tel.: +86 29 884 95 453; fax: +86 29 884 95 453. E-mail addresses: weixu@nwpu.edu.cn (W. Xu), liwei@mail.nwpu.edu.cn (W. Li).