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
Consensus and synchronization in discrete-time networks of multi-agents with stochastically switching topologies and time delays
Wenlian Lu Fatihcan M. Atay Jürgen Jost
Networks & Heterogeneous Media 2011, 6(2): 329-349 doi: 10.3934/nhm.2011.6.329
We analyze stability of consensus algorithms in networks of multi-agents with time-varying topologies and delays. The topology and delays are modeled as induced by an adapted process and are rather general, including i.i.d. topology processes, asynchronous consensus algorithms, and Markovian jumping switching. In case the self-links are instantaneous, we prove that the network reaches consensus for all bounded delays if the graph corresponding to the conditional expectation of the coupling matrix sum across a finite time interval has a spanning tree almost surely. Moreover, when self-links are also delayed and when the delays satisfy certain integer patterns, we observe and prove that the algorithm may not reach consensus but instead synchronize at a periodic trajectory, whose period depends on the delay pattern. We also give a brief discussion on the dynamics in the absence of self-links.
keywords: Consensus network of multi-agents synchronization adapted process switching topology. delay

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