Coupled Complex Dynamical Networks Research Articles

Robust global synchronization of two complex dynamical networks

We investigate the synchronization of two coupled complex dynamical networks, a problem that has been termed outer synchronization in the literature. Our approach relies on (a) a basic lemma on the eigendecomposition of matrices resulting from Kronecker products and (b) a suitable choice of Lyapunov function related to the synchronization error dynamics. Starting from these two ingredients, a theorem that provides a sufficient condition for outer synchronization of the networks is proved. The condition in the theorem is expressed as a linear matrix inequality. When satisfied, synchronization is guaranteed to occur globally, i.e., independently of the initial conditions of the networks. The argument of the proof includes the design of the gain of the synchronizer, which is a constant square matrix with dimension dependent on the number of dynamic variables in a single network node, but independent of the size of the overall network, which can be much larger. This basic result is subsequently elaborated to simplify the design of the synchronizer, to avoid unnecessarily restrictive assumptions (e.g., diffusivity) on the coupling matrix that defines the topology of the networks and, finally, to obtain synchronizers that are robust to model errors in the parameters of the coupled networks. An illustrative numerical example for the outer synchronization of two networks of classical Lorenz nodes with perturbed parameters is presented.

Topology identification of uncertain nonlinearly coupled complex networks with delays based on anticipatory synchronization

This paper presents an adaptive anticipatory synchronization based method for simultaneous identification of topology and parameters of uncertain nonlinearly coupled complex dynamical networks with time delays. An adaptive controller is proposed, based on Lyapunov stability theorem and Barbǎlat's Lemma, to guarantee the stability of the anticipatory synchronization manifold between drive and response networks. Meanwhile, not only the identification criteria of network topology and system parameters are obtained but also the anticipatory time is identified. Numerical simulation results illustrate the effectiveness of the proposed method.

CHAOS GENERATION FOR A CLASS OF NONLINEAR COMPLEX NETWORKS

Chaos generation is an interesting research topic in the study of coupled complex dynamical networks. In this paper, based on mathematical analysis of Lyapunov exponent and boundedness of networks, the emergence of chaos for a class of nonlinear complex networks is investigated and some new criteria of chaos generation are derived. The effectiveness of theoretical results is verified by a numerical example.

Cluster projective synchronization of complex networks with nonidentical dynamical nodes

We investigate a new cluster projective synchronization (CPS) scheme in time-varying delay coupled complex dynamical networks with nonidentical nodes. Based on the community structure of the networks, the controllers are designed differently for the nodes in one community, which have direct connections to the nodes in the other communities and the nodes without direct connections to the nodes in the other communities. Some sufficient criteria are derived to ensure the nodes in the same group projectively synchronize and there is also projective synchronization between nodes in different groups. Particularly, the weight configuration matrix is not assumed to be symmetric or irreducible. The numerical simulations are performed to verify the effectiveness of the theoretical results.

Effects of noise on the outer synchronization of two unidirectionally coupled complex dynamical networks

We study the effect of noise on the outer synchronization between two unidirectionally coupled complex networks and find analytically that outer synchronization could be achieved via white-noise-based coupling. It is also demonstrated that, if two networks have both conventional linear coupling and white-noise-based coupling, the critical deterministic coupling strength between two complex networks for synchronization transition decreases with an increase in the intensity of noise. We provide numerical results to illustrate the feasibility and effectiveness of the theoretical results.

Hybrid synchronization of the general delayed and non-delayed complex dynamical networks via pinning control

This paper investigates the hybrid synchronization problem of two coupled complex dynamical networks with non-delayed and delayed coupling by the pinning control strategy. Based on the LaSalle invariance principle and linear matrix inequality technique, we obtain some sufficient conditions for the hybrid synchronization by applying the simple linear feedback and adaptive controllers to a part of nodes. Under suitable conditions, two coupled networks can reach the hybrid synchronization, i.e., the outer synchronization between the drive and response networks, and the inner synchronization in each network simultaneously. Numerical simulations show the effectiveness of the proposed synchronization scheme.

Cluster Synchronization of Time‐Varying Delays Coupled Complex Networkswith Nonidentical Dynamical Nodes

This paper investigates a new cluster synchronization scheme in the nonlinear coupled complex dynamical networks with nonidentical nodes. The controllers are designed based on the community structure of the networks; some sufficient criteria are derived to ensure cluster synchronization of the network model. Particularly, the weight configuration matrix is not assumed to be symmetric, irreducible. The numerical simulations are performed to verify the effectiveness of the theoretical results.

Adaptive‐Impulsive Control of the Projective Synchronization in Drive‐Response Complex Dynamical Networks with Time‐Varying Coupling

This paper investigates the projective synchronization (PS) of drive‐response time‐varying coupling complex dynamical networks with time delay via an adaptive‐impulsive controlling method, in which the weights of links are time varying. Based on the stability analysis of impulsive control system, sufficient conditions for the PS are derived, and a hybrid controller, that is, an adaptive feedback controller with impulsive control effects, is designed. Numerical simulations are performed to verify the correctness and effectiveness of theoretical result.

Adaptive synchronization of two nonlinearly coupled complex dynamical networks with delayed coupling

This paper investigates the adaptive synchronization between two nonlinearly delay-coupled complex networks with the bidirectional actions and nonidentical topological structures. Based on LaSalle’s invariance principle, some criteria for the synchronization between two coupled complex networks are achieved via adaptive control. To validate the proposed methods, the unified chaotic system as the nodes of the networks are analyzed in detail, and numerical simulations are given to illustrate the theoretical results.

Exponential synchronization of weighted general delay coupled and non-delay coupled dynamical networks

Time delays commonly exist in the real world. In the present work, we investigate the exponential synchronization of weighted general delay coupled and non-delay coupled complex dynamical networks with different topological structures. Based on the Lyapunov stability theory, the suitable controllers are designed to make the controlled dynamical network exponentially synchronize an isolated node with any pre-specified exponential convergence rate, and proved theoretically. The synchronization scheme is applicable to the undirected networks as well as the directed ones. The derived controllers are simple and can be readily used in practical applications. Furthermore, the coupling matrix is not necessary to be irreducible and the network node dynamics need not satisfy the very strong and conservative uniform Lipschitz condition. Numerical simulations further validate the effectiveness and feasibility of our synchronization method.