On transverse stability of random dynamical system
Xiangnan He Wenlian Lu Tianping Chen
Discrete & Continuous Dynamical Systems - A 2013, 33(2): 701-721 doi: 10.3934/dcds.2013.33.701
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.
keywords: RDS stochastic topologies and maps Lyapunov exponents complete synchronization. transverse stability
Cluster synchronization for linearly coupled complex networks
Xiwei Liu Tianping Chen Wenlian Lu
Journal of Industrial & Management Optimization 2011, 7(1): 87-101 doi: 10.3934/jimo.2011.7.87
In this paper, the cluster synchronization for an array of linearly coupled identical chaotic systems is investigated. New coupling schemes (or coupling matrices) are proposed, by which global cluster synchronization of linearly coupled chaotic systems can be realized. Here, the number and the size of clusters (or groups) can be arbitrary. Some sufficient criteria to ensure global cluster synchronization are derived. Moreover, for any given coupling matrix, new coupled complex networks with adaptive coupling strengths are proposed, which can synchronize coupled chaotic systems by clusters. Numerical simulations are finally given to show the validity of the theoretical results.
keywords: adaptive coupling strengths. complex networks Cluster synchronization left eigenvector

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