Positive role of multiplication noise in attaining complete synchronization on large complex networks of dynamical systems

In this paper, the role of multiplicative noise in attaining complete synchronization on large complex networks of dynamical systems is investigated by theoretical analysis and numerical simulations. Based on the stability theory of stochastic differential equation, we prove that the multiplicative noise plays a positive role in attaining synchronization if the complex networks of dynamical systems are bounded. Moreover, the theoretical result shows that smaller second eigenvalue of coupling matrix is of benefit in attaining complete synchronization. To demonstrate the correctness of theoretical results, the coupled Lorenz systems, Hindmarsh–Rose neuronal systems and Rössler-like systems are performed as numerical examples.     

Synchronization of networks with time-varying couplings

In this paper, we present a review of our recent works on complete synchro-nization analyses of networks of the coupled dynamical systems with time-varying cou-plings. The main approach is composed of algebraic graph theory and dynamic system method. More precisely, the Hajnal diameter of matrix sequence plays a key role in in-vestigating synchronization dynamics and the joint graph across time periods possessing spanning tree is a doorsill for time-varying topologies to reach synchronization. These techniques with proper modification count for diverse models of networks of the cou-pled systems, including discrete-time and continuous-time models, linear and nonlinear models, deterministic and stochastic time-variations. Alternatively, transverse stability analysis of general time-varying dynamic systems can be employed for synchronization study as a special case and proved to be equivalent to Hajnal diameter.     

Impulsive synchronization seeking in general complex delayed dynamical networks

The present paper investigates the issues of impulsive synchronization seeking in general complex delayed dynamical networks with nonsymmetrical coupling. By establishing the extended Halanay differential inequality on impulsive delayed dynamical systems, some simple yet generic sufficient conditions for global exponential synchronization of the impulsive controlled delayed dynamical networks are derived analytically. Compared with some existing works, the distinctive features of these sufficient conditions indicate two aspects: on the one hand, these sufficient conditions can provide an effective impulsive control scheme to synchronize an arbitrary given delayed dynamical network to a desired synchronization state even if the original given network may be asynchronous itself. On the other hand, the controlled synchronization state can be selected as a weighted average of all the states in the network for the purpose of practical control strategy, which reveals the contributions and influences of various nodes in synchronization seeking processes of the dynamical networks. It is shown that impulses play an important role in making the delayed dynamical networks globally exponentially synchronized. Furthermore, the results are applied to a typical nearest-neighbor unidirectional time-delay coupled networks composed of chaotic FHN neuron oscillators, and numerical simulations are given to demonstrate the effectiveness of the proposed control methodology.     

Synchronization of systems of Marcus canonical equations driven by α-stable noises

Dynamical systems driven by Gaussian noises have been considered extensively in modeling, simulation and theory. However, complex systems in engineering and science are often subject to non-Gaussian fluctuations or uncertainties. A coupled dynamical system under non-Gaussian Lévy noise is considered. After discussing cocycle property, stationary orbits and random attractors, a synchronization phenomenon is shown to occur, when the drift terms of the coupled system satisfy certain dissipativity conditions. The synchronization result implies that coupled dynamical systems share a dynamical feature in certain asymptotic sense.     

Adaptive synchronization of bipartite dynamical networks with distributed delays and nonlinear derivative coupling

This paper investigates the synchronization in a class of bipartite dynamical networks with distributed delays and nonlinear derivative coupling. Based on Lyapunov stability theory, some useful synchronization criteria are established for the two coupled bipartite dynamical networks by constructing effective adaptive feedback controllers and update laws. The numerical simulations are provided to illustrate the effectiveness of the theoretical results obtained in this paper.     

脉冲控制下复杂网络的同步

主要考虑非对称耦合复杂网络的脉冲同步问题.通过构造Lyapunov泛函,设计合适的脉冲控制器,并利用时滞脉冲系统理论,给出了网络脉冲同步新的判别准则.数值模拟表明所得结果是正确的.     

Striped attractor generation and synchronization analysis for coupled Rössler systems

A model of networked chaotic Rössler systems with periodic couplings is discussed. New phenomena, including individual attractors in striped rectangular shapes and partial synchronization (or clustering), are shown for these locally coupled systems. Coupling-induced attractors with multiple stripes can be easily controlled by coupling parameters. Moreover, various interconnection topologies are also taken into consideration in the synchronization analysis, and dynamical behaviors of the coupled systems are illustrated by numerical results.     

A new synchronization algorithm for delayed complex dynamical networks via adaptive control approach

An adaptive control strategy is developed for complex delayed dynamical networks with time-varying coupling strength and time-varying delayed. Using the Lyapunov stability theory, an adaptive control is designed to ensure the asymptotic convergence of the synchronization error, a sufficient condition of the synchronization is obtained. By constructing a Lyapunov–Krasovskii-like composite energy function, we prove the stability of the closed-loop system and the convergence of the error. An example of the complex network is finally used to verify the proposed theoretical result.     

Robust adaptive synchronization of uncertain complex networks with multiple time‐varying coupled delays

In this article, the synchronization problem of uncertain complex networks with multiple coupled time‐varying delays is studied. The synchronization criterion is deduced for complex dynamical networks with multiple different time‐varying coupling delays and uncertainties, based on Lyapunov stability theory and robust adaptive principle. By designing suitable robust adaptive synchronization controllers that have strong robustness against the uncertainties in coupling matrices, the all nodes states of complex networks globally asymptotically synchronize to a desired synchronization state. The numerical simulations are given to show the feasibility and effectiveness of theoretical results. © 2014 Wiley Periodicals, Inc. Complexity 20: 62–73, 2015     

Observer-based lag synchronization between two different complex networks

In this paper, some new criteria for lag synchronization between two or more complex networks are proposed based on the theory of state observer. Some adaptive controllers are designed to make the drive and response systems achieve lag synchronization, no matter whether the nodes in the two systems are with the same dynamical character or the coupling configuration matrices are nonidentical. In addition, based on the output coupling, the amount of coupling variables between two connected nodes is flexible, which can save a lot of channel resources, simplify the network topology and has more significant meanings in engineering applications. At last, the effects of the lag synchronization criteria are verified through some simulation experiments.     

复杂动态网络的有限时间同步

复杂网络无处不在,同步是自然界中广泛存在的一类非常重要的非线性现象.过去10年,人们对复杂网络的同步开展了系统而深入的研究,包括恒等同步、广义同步、簇同步以及部分同步等.上述大部分结果中对同步速度的刻画往往是渐进的,只有当时间趋于无穷的时候,网络才能实现同步,而对于网络能够在多长时间内可以实现同步却知之甚少.作者以几类典型的非线性耦合的复杂动态网络为例,深入探讨了复杂动态网络的有限时间同步的规律.具体而言,基于上述几类典型的复杂动态网络,证明了在某些合适的条件下,网络能够在有限时间内实现精确同步.此外,用一个典型的数值仿真实例验证了上述有限时间同步的准则.有限时间同步有效地避免了网络只有在无穷时刻才能实现同步的问题,对网络同步的实际工程应用具有基本的现实意义.     

Universal occurrence of mixed-synchronization in counter-rotating nonlinear coupled oscillators

By coupling counter-rotating coupled nonlinear oscillators, we observe a “mixed” synchronization between the different dynamical variables of the same system. The phenomenon of amplitude death is also observed. Results for coupled systems with co-rotating coupled oscillators are also presented for a detailed comparison. Results for Landau–Stuart and Rössler oscillators are presented.     

The wavelet transform method applied to a coupled chaotic system with asymmetric coupling schemes

The wavelet transform method originated by Wei et al. (2002) [19] is an effective tool for enhancing the transverse stability of the synchronous manifold of a coupled chaotic system. Much of the theoretical study on this matter is centered on networks that are symmetrically coupled. However, in real applications, the coupling topology of a network is often asymmetric; see Belykh et al. (2006)  [23], [24], Chavez et al. (2005)  [25], Hwang et al. (2005)  [26], Juang et al. (2007)  [17], and Wu (2003)  [13]. In this work, a certain type of asymmetric sparse connection topology for networks of coupled chaotic systems is presented. Moreover, our work here represents the first step in understanding how to actually control the stability of global synchronization from dynamical chaos for asymmetrically connected networks of coupled chaotic systems via the wavelet transform method. In particular, we obtain the following results. First, it is shown that the lower bound for achieving synchrony of the coupled chaotic system with the wavelet transform method is independent of the number of nodes. Second, we demonstrate that the wavelet transform method as applied to networks of coupled chaotic systems is even more effective and controllable for asymmetric coupling schemes as compared to the symmetric cases.     

Function projective synchronization in complex dynamical networks with time delay via hybrid feedback control

This paper investigates the problem of function projective synchronization for general complex dynamical networks with time delay. A hybrid feedback control method is designed to achieve function projective synchronization for complex dynamical networks, one with constant time delay and one with time-varying coupling delay. Numerical examples are provided to show the effectiveness of the proposed method.     

Reduced-order synchronization of uncertain chaotic systems via adaptive control

We consider the coupling of two uncertain dynamical systems with different orders using an adaptive feedback linearization controller to achieve reduced-order synchronization between the two systems. Reduced-order synchronization is the problem of synchronization of a slave system with projection of a master system. The synchronization scheme is an exponential linearizing-like controller and a state/uncertainty estimator. As an illustrative example, we show that the dynamical evolution of a second-order driven oscillator can be synchronized with the canonical projection of a fourth-order chaotic system. Simulation results indicated that the proposed control scheme can significantly improve the synchronousness performance. These promising results justify the usefulness of the proposed output feedback controller in the application of secure communication.     

Exponential Synchronization of the Linearly Coupled Dynamical Networks with Delays

In this paper,the authors investigate the synchronization of an array of linearly coupled identical dynamical systems with a delayed coupling.Here the coupling matrix can be asymmetric and reducible.Some criteria ensuring delay-independent and delay- dependent global synchronization are derived respectively.It is shown that if the coupling delay is less than a positive threshold,then the coupled network will be synchronized.On the other hand,with the increase of coupling delay,the synchronization stability of the network will be restrained,even eventually de-synchronized.     

Generalized function projective (lag, anticipated and complete) synchronization between two different complex networks with nonidentical nodes

Generalized function projective (lag, anticipated and complete) synchronization between two different complex networks with nonidentical nodes is investigated in this paper. Based on Barbalat’s lemma, some sufficient synchronization criteria are derived by applying the nonlinear feedback control. Although previous work studied function projective synchronization on complex dynamical networks, the dynamics of the nodes are coupled partially linear chaotic systems. In our work, the dynamics of the nodes of the complex networks are any chaotic systems without the limitation of the partial linearity. In addition, each network can be undirected or directed, connected or disconnected, and nodes in either network may have identical or different dynamics. The proposed strategy is applicable to almost all kinds of complex networks. Numerical simulations further verify the effectiveness and feasibility of the proposed synchronization method. Numeric evidence shows that the synchronization rate is sensitively influenced by the feedback strength, the time delay, the network size and the network topological structure.     

Network synchronization under distributed delayed impulsive control: Average delayed impulsive weight approach

This paper mainly investigates synchronization of complex dynamical networks (CDNs) with both system delay and coupled delay through distributed delayed impulsive control. Instead of constraining the impulsive weight and impulsive delay one by one, a new concept of average delayed impulsive weight is proposed to obtain more relaxed conditions. Subsequently, based on the impulsive control topology, Lyapunov theory and linear matrix inequality (LMI) design, certain flexible criteria of global exponential synchronization (GES) are given and the corresponding convergence rate is estimated. It is interesting to see that the CDNs can still achieve synchronization under comprehensive conditions though impulsive weights work negatively. Namely, the delays in impulsive control are able to promote synchronization potentially. Finally, simulations are given to show that the distributed delayed impulsive control can not only speeds up the convergence rate for synchronized networks, but also facilitates synchronization for desynchronized networks. In addition, the obtained results can be applied to unmanned craft systems.     

Generalized outer synchronization between non-dissipatively coupled complex networks with different-dimensional nodes

This paper investigates the generalized outer synchronization (GOS) between two non-dissipatively coupled complex dynamical networks (CDNs) with different time-varying coupling delays. Our drive-response networks also possess nonlinear inner coupling functions and time-varying outer coupling configuration matrices. Besides, in our network models, the nodes in the same network are nonidentical and the nodes in different networks have different state dimensions. Asymptotic generalized outer synchronization (AGOS) and exponential generalized outer synchronization (EGOS) are defined for our CDNs. Our main objective in this paper is to design AGOS and EGOS controllers for our drive-response networks via the open-plus-closed-loop control technique. Distinguished from most existing literatures, it is the partial intrinsic dynamics of each node in response network that is restricted by the QUAD condition, which is easy to be satisfied. Representative simulation examples are given to verify the effectiveness and feasibility of our GOS theoretical results in this paper.     

Synchronization of neural networks based on parameter identification and via output or state coupling

For neural networks with all the parameters unknown, we focus on the global robust synchronization between two coupled neural networks with time-varying delay that are linearly and unidirectionally coupled. First, we use Lyapunov functionals to establish general theoretical conditions for designing the coupling matrix. Neither symmetry nor negative (positive) definiteness of the coupling matrix are required; under less restrictive conditions, the two coupled chaotic neural networks can achieve global robust synchronization regardless of their initial states. Second, by employing the invariance principle of functional differential equations, a simple, analytical, and rigorous adaptive feedback scheme is proposed for the robust synchronization of almost all kinds of coupled neural networks with time-varying delay based on the parameter identification of uncertain delayed neural networks. Finally, numerical simulations validate the effectiveness and feasibility of the proposed technique.