Invariant foliations are geometric structures for describing and understanding the qualitative behaviors of nonlinear dynamical systems. For stochastic dynamical systems, however, these geometric structures themselves are complicated random sets. Thus it is desirable to have some techniques to approximate random invariant foliations. In this paper, invariant foliations are approximated for dynamical systems with small noisy perturbations, via asymptotic analysis. Namely, random invariant foliations are represented as a perturbation of the deterministic invariant foliations, with deviation errors estimated.
Bibliographical noteFunding Information:
Part of this work was done while J. Duan was participating the Stochastic Partial Differential Equations program at the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK. This work was partly supported by NSF of China grants 10971225 and 11028102, the NSF Grants 1025422 and 0731201, the Fundamental Research Funds for the Central Universities-HUST 2010ZD037, and an open research grant from the State Key Laboratory for Nonlinear Mechanics at the Chinese Academy of Sciences.
- Stable and unstable foliations
- asymptotic expansion
- fiber or leaf
- random dynamical systems