Study On Synchronization Of The Complex Chaotic Dynamical Network Systems And Its Applications

Complexity and complex systems are the important topics in the 21st century. Complex networks are presently significant tools and methods to describe and understand the complex system, which highly summarize the complex system as the networks consisting of many interacting individuals or nodes. Synchronization of complex dynamical networks is one of significant topic in complex networks, which has great potential applications in secure communication, network congestion control, the generation of harmonic oscillator, multi-agent consensus, and so forth. The research on the nonlinear dynamics, especially on chaos and chaotic synchronization, are used as theoretical basis and tools for studying synchronization of complex networks.In this dissertation, based on the research on synchronization of (fractional-order) hyperchaotic systems and its application in secure communication, different types of complex networks, i.e., the fractional-order chaotic dynamical networks, the general chaotic dynamical networks with delay coupling and the time-varying chaotic dynamical networks, are investigated. The effective controllers for complex dynamical networks are designed to guarantee to achieve the desired network synchronization. The main contributions of this dissertation are summarized as follows:(1) Generalized function projective lag synchronization of uncertain hyperchaotic systems and its application in secure communication are studied. When the parameters of the drive and response systems are fully unknown or only the parameters of the response system are unknown, based on Lyapunov stability theory and the adaptive control method, two different universal adaptive generalized function projective lag synchronization schemes are proposed. Both rigorous theoretical proofs and numerical simulations are given to validate the effectiveness and robustness of the proposed synchronization methods. The influence of the scaling factor and time delay on the synchronization effect is further discussed. On the basis of the above studies, combining the parameter modulation and chaotic masking techniques, two distinct hyperchaotic secure communication schemes are proposed by applying generalized function projective synchronization of Chen hyperchaotic system. Corresponding theoretical proofs and numerical simulations demonstrate the validity and feasibility of the presented hyperchaotic secure communication schemes.(2) The problem of the dynamical properties and synchronization of the fractional-order hyperchaotic systems is investigated. Firstly, based on the fractional calculus theory and computer simulations, the dynamical properties of two new fractional-order four-dimensional systems are analyzed. The lowest orders to have hyperchaos in two systems are also obtained, respectively. Secondly, based on the stability theory of the fractional-order systems, by using the state observer method, the active control method and the system coupling method, three general fractional-order chaos synchronization methods are presented. Furthermore, by applying generalized projective synchronization of the fractional-order hyperchaotic system and chaotic masking technique, a fractional-order hyperchaotic secure communication scheme is constructed, which has a larger key space and is more secure than the integer-order chaotic communication. Finally, a general method for parameter identification and synchronization of uncertain fractional-order chaotic systems is proposed and proved theoretically based on the stability theory of the fractional-order systems. The simulation results are performed to verify the validity of the presented method.(3) Synchronization of the fractional-order complex chaotic dynamical networks is considered. Firstly, the outer synchronization between two fractional-order chaotic dynamical networks with identical or different topology is studied by applying the nonlinear control and bidirectional coupling methods. The sufficient criteria for the outer synchronization are derived analytically. Numerical results show that the larger the fractional order and the feedback gain, the faster is to achieve the outer synchronization; it is much easier to realize the outer synchronization between two networks with identical topology and uniform node dynamics. Secondly, the nonlinear controllers are designed to realize generalized synchronization of the fractional-order chaotic dynamical networks with distinct nodes, and some sufficient synchronization criteria are obtained. Simulation results show that the synchronization speed sensitively depends on both the fractional order and the feedback gain. For the same feedback gain, the synchronization effect of the integer-order dynamical network is much better than that of the fractional-order one. Furthermore, by the nonlinear controllers, the network synchronization can still be achieved effectively in presence of noise and parameter perturbations.(4) We study synchronization of complex chaotic dynamical networks with delayed coupling. Firstly, a general complex dynamical network model with non-delayed and delayed coupling is proposed. The nodes in the network may be coupled nonlinearly or linearly. And the network can be directed or undirected. Only partial information of the coupling configuration matrix is used to design the controllers to achieve the exponential synchronization of the dynamical networks with non-delayed and delayed coupling. Both theoretical analysis and numerical simulations have validated the effectiveness of the synchronization method. Secondly, based on the LaSalle invariant principle and adaptive control method, generalized projective synchronization between two completely different complex dynamical networks with delayed coupling can be obtained by constructing the adaptive controllers. Corresponding theoretical proofs are also given. Numerical simulations further demonstrate the validity of the theoretical results.(5) Synchronization of the time-varying complex chaotic dynamical networks is investigated. A general complex dynamical network model with adaptive coupling strengths and community structure is firstly proposed. The local dynamics of individual nodes in each community are identical, while those of any pair of nodes in different communities are diverse. The local controllers and the adaptive law for the coupling strengths are designed to achieve cluster synchronization in the adaptive dynamical networks. We take the BA scale-free network and the WS small-world network as the examples to do the simulations. Numeric evidences show that the synchronization performance is sensitively affected by the network topological structure, the inner-coupling matrix, the rewiring probability and the control gain; the BA scale-free network is much easier to achieve cluster synchronization than the WS small-world network; the presented scheme is robust against noise. Secondly, we present a general delayed dynamical network model with different nodes and time-varying coupling configuration matrix. The adaptive controllers and corresponding parameter update rule are constructed to achieve the outer synchronization and parameter identification of uncertain time-varying delayed dynamical networks. Theoretical analysis and numerical simulations have verified the effectiveness of the proposed scheme.(6) The hybrid synchronization problem of two coupled complex dynamical networks with non-delayed and delayed coupling is investigated by the pinning control strategy. Based on the LaSalle invariance principle and linear matrix inequality technique, we obtain some sufficient synchronization conditions by applying the simple linear feedback controllers and adaptive controllers to a part of nodes, respectively. Numerical results show that the hybrid synchronization of two coupled networks can be realized by a single controller; the synchronization effect turns better with the decrease of time delay; the hybrid synchronization is easier to realize with low cost by the adaptive control method than that by the linear feedback control approach.