On transverse stability of random dynamicalsystem

Xiangnan He; Wenlian Lu; Tianping Chen

doi:10.3934/dcds.2013.33.701

Article Contents

2013, Volume 33, Issue 2: 701-721. Doi: 10.3934/dcds.2013.33.701

This issue Previous Article Solitary waves of the rotation-generalized Benjamin-Ono equation Next Article On global existence of classical solutions for the Vlasov-Poisson system in convex bounded domains

On transverse stability of random dynamical system

1.
Center for Computational Systems Biology, Laboratory of Mathematics for Nonlinear Sciences, School of Mathematical Sciences, Fudan University，Shanghai, 200433
2.
Center for Computational Systems Biology, Laboratory of Mathematics for Nonlinear Sciences, School of Mathematical Sciences, Fudan University, Shanghai, 200433
3.
Laboratory of Mathematics for Nonlinear Sciences, School of Mathematical Sciences, Fudan University, Shanghai, 200433

Received: June 30, 2011

Revised: March 31, 2012

Published: January 2013

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Abstract

In this paper, we study the transverse stability of random dynamical systems (RDS). Suppose a RDS on a Riemann manifold possesses a non-random invariant submanifold, what conditions can guarantee that a random attractor of the RDS restrained on the invariant submanifold is a random attractor with respect to the whole manifold? By the linearization technique, we prove that if all the normal Lyapunov exponents with respect to the tangent space of the submanifold are negative, then the attractor on the submanifold is also a random attractor of the whole manifold. This result extends the idea of the transverse stability analysis of deterministic dynamical systems in [1,3]. As an explicit example, we discuss the complete synchronization in network of coupled maps with both stochastic topologies and maps, which extends the well-known master stability function (MSF) approach for deterministic cases to stochastic cases.

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Mathematics Subject Classification: Primary: 37H15, 37D10; Secondary: 60F10.

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