WIAS Preprint No. 647, (2001)

The moment Lyapunov exponent for conservative systems with small periodic and random perturbations


  • Imkeller, Peter
  • Milstein, Grigori N.

2010 Mathematics Subject Classification

  • 60H10 93E15


  • Linear stochastic systems with periodic coefficients, stochastic stability, moment Lyapunov exponent, stability index, Hill and Mathieu equations with random excitations




Much effort has been devoted to the stability analysis of stationary points for linear autonomous systems of stochastic differential equations. Here we introduce the notions of Lyapunov exponent, moment Lyapunov exponent, and stability index for linear nonautonomous systems with periodic coefficients. Most extensively we study these problems for second order conservative systems with small random and periodic excitations. With respect to relations between the intrinsic period of the system and the period of perturbations we consider the incommensurable and commensurable cases. In the first case we obtain an asymptotic expansion of the moment Lyapunov exponent. In the second case we obtain a finite expansion except in situations of resonance. As an application we consider the Hill and Mathieu equations with random excitations.

Appeared in

  • Stochastics and Dynamics, vo. 2 (2002), no. 1, pp. 25-48

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