WIAS Preprint No. 244, (1996)

Stability Index for Invariant Manifolds of Stochastic Systems


  • Milstein, Grigori N.

2010 Mathematics Subject Classification

  • 60H10 93E15 34C30 34D08 34F05


  • stochastic stability, Lyapunov exponents, moment Lyapunov exponents, stability index


A lot of works has been devoted to stability analysis of a stationary point for linear and non-linear systems of stochastic differential equations. Here we consider the stability of an invariant compact manifold of a non-linear system. To this end we derive a linearized system for orthogonal displacements of a solution from the manifold. For this system, we introduce notions of Lyapunov exponents, moment Lyapunov exponents, and stability index. The stability index controls the asymptotic behavior of solutions of the input system in a neighborhood of the manifold. Most extensively we study these problems in the case when the invariant manifold is an orbit.

Appeared in

  • Random Comput. Dynamics 5 (1997) no. 4

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