Adaptive synchronization in tree-like dynamical networks

Synchronization in an array of coupled identical nonlinear dynamical systems have attracted increasing attention from various fields of science and engineering. In this paper, we investigate the synchronization phenomenon in tree-like dynamical networks. Based on the LaSalle invariant principle, a simple and systematic adaptive control scheme with variable coupling strength is proposed for the synchronization of tree-like dynamical networks without any knowledge of the concrete structure of isolate system. This result indicates that synchronization can be achieved for strong enough coupling if there exists a system (located at the root of the tree) which directly or indirectly influences all other systems. Furthermore, the main result is applied to several Lorenz chaotic systems coupled by a tree. And numerical simulations are also given to show the effectiveness of the proposed synchronization method.     

A novel criterion for cluster synchronization of complex dynamical networks with coupling time-varying delays

This paper investigates the cluster synchronization problem for the time-varying delays coupling networks with nonidentical delayed dynamical systems by using pinning control method. We derive some simple and useful criteria for cluster synchronization for any initial values through an effective feedback control scheme and propose an adaptive feedback strategy that adjusts automatically the coupling strength. Finally, some numerical examples are then given to illustrate the theoretical results.     

Synchronization for fractional‐order multi‐linked complex network with two kinds of topological structure via periodically intermittent control

This paper concentrates on the global synchronization of the fractional‐order multi‐linked complex network (FMCN) via periodically intermittent control. It should be stressed that periodically intermittent control is employed to the FMCN for the first time. Moreover, the network is defined on digraphs with different weights, and two situations on topological structure of the network are discussed, including each digraph being strongly connected, and the biggest one being strongly connected. Based on Lyapunov method and graph theory, some synchronization criteria are obtained under two situations. And, the obtained synchronization criteria have a close relationship with the order of fractional‐order derivative, coupling strength, control gain, control rate, and control period. Besides, for practicability, theoretical results are applied to studying the synchronization of fractional‐order multi‐linked chaotic systems, and some sufficient conditions are provided. For a special case, fractional‐order multi‐linked Lorenz chaotic systems, numerical simulations are given to indicate the feasibility of theoretical results and the effectiveness of control strategy.     

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.     

Feedback control based on discrete-time state observations on synchronization of stochastic impulsive coupled systems

In this paper, feedback control based on discrete-time state observations is used to study the inner synchronization of stochastic impulsive coupled systems (SICSs). Therein, the coupling strength of SICSs is state-dependent switching and time-varying under each switching. Besides, by means of average impulsive interval approach, the Lyapunov method and the graph theory, a synchronization criterion of SICSs is presented. As an application, stochastic impulsive coupled Chua’s circuits with state-dependent switching coupling strength are investigated for the first time and some sufficient conditions are given. Finally, in order to illustrate the effectiveness of our main results, some numerical simulations are presented.     

Projective synchronization with different scale factors in a driven–response complex network and its application in image encryption

We study projective (anticipatory, exact and lag) synchronization of time-delayed chaotic systems in a driven–response complex network (DRCN), where the nodes are not partially linear and the scale factors are different from each other. Two synchronization criteria are presented; the first is easy to satisfy with simple limitation and the second relaxes the previous limitation and can be executed only by adding a simple linear feedback control to every node. As a special case, projective synchronization in a DRCN without time delay is analyzed by using a different method. The corresponding synchronization criterion is given, which can be realized by adopting single-node adaptive control. This paper extends previous work, where the nodes of the DRCN are partially linear, there is limitation of the coupling matrix or the scale factors of the nodes are all equal. Numerical examples are given to demonstrate the effectiveness of the proposed synchronization criteria. Due to the complexity of the different scale factors, we select the chaotic data from the DRCN presented in this paper and apply the data selected in image encryption. Simulation results show that our method is efficient with a large key space, high sensitivity to initial conditions and high speed.     

Synchronization of van der Pol oscillator and Chen chaotic dynamical system

This paper addresses the synchronization problem of two different electronic circuits by using nonlinear control function. This technique is applied to achieve synchronization for the stable van der Pol oscillator and Chen chaotic dynamical system. Numerical simulations results are given to demonstrate the effectiveness of the proposed control method.     

Finite-time synchronization control of complex dynamical networks with time delay

In this paper, the finite-time synchronization between two complex networks with non-delayed and delayed coupling is proposed by using the impulsive control and the periodically intermittent control. Some novel and useful finite-time synchronization criteria are derived based on finite-time stability theory. Especially, the traditional synchronization criteria are improved by using the impulsive control and the periodically intermittent control in the convergence time, the results of this paper are important. Finally, numerical examples are given to verify the effectiveness and correctness of the synchronization criteria.     

Synchronization of quaternion-valued coupled systems with time-varying coupling via event-triggered impulsive control

In this paper, the problem of exponential synchronization of quaternion-valued coupled systems based on event-triggered impulsive control is investigated for the first time. It should be pointed out that the coupling strength is quaternion-valued and time-varying, which makes our model more in line with practical models. First, we prove that event-triggered impulsive control can exclude Zeno behavior. Then, based on the Lyapunov method and the graph theory, some sufficient conditions are derived to ensure that quaternion-valued coupled systems reach synchronization. Furthermore, as an application of our theoretical results, exponential synchronization of quaternion-valued Kuramoto oscillators is studied in detail and a synchronization criterion is presented. Finally, some numerical simulations are given to show the effectiveness of our theoretical results.     

Synchronization of fractional‐order delayed neural networks with hybrid coupling

This article presents new theoretical results on the synchronization for a class of fractional‐order delayed neural networks with hybrid coupling that contains constant coupling and discrete‐delay coupling. This is the first attempt to investigate the synchronization problem of fractional‐order coupled delayed neural networks. Based on the fractional‐order Lyapunov stability theorem and Kronecker product properties, sufficient criteria are established to ensure the fractional‐order coupled neural network to achieve synchronization. Numerical simulations are given to illustrate the correctness of the theoretical results. © 2015 Wiley Periodicals, Inc. Complexity 21: 106–112, 2016     

Outer synchronization between two hybrid‐coupled delayed dynamical networks via aperiodically adaptive intermittent pinning control

This article is concerned with the problem of pinning outer synchronization between two complex delayed dynamical networks via adaptive intermittent control. At first, a general model of hybrid‐coupled dynamical network with time‐varying internal delay and time‐varying coupling delay is given. Then, an aperiodically adaptive intermittent pinning‐control strategy is introduced to drive two such delayed dynamical networks to achieve outer synchronization. Some sufficient conditions to guarantee global outer‐synchronization are derived by constructing a novel piecewise Lyapunov function and utilizing stability analytical method. Moreover, a simple pinned‐node selection scheme determining what kinds of nodes should be pinned first is provided. It is noted that the adaptive pinning control type is aperiodically intermittent, where both control period and control width are non‐fixed. Finally, a numerical example is given to illustrate the validity of the theoretical results. © 2016 Wiley Periodicals, Inc. Complexity 21: 593–605, 2016     

Synchronization of complex chaotic systems in series expansion form

This paper studies the synchronization of complex chaotic systems in series expansion form by Lyapunov asymptotical stability theorem. A sufficient condition is given for the asymptotical stability of an error dynamics, and is applied to guiding the design of the secure communication. Finally, numerical results are studied for the Quantum-CNN oscillators synchronizing with unidirectional/bidirectional linear coupling to show the effectiveness of the proposed synchronization strategy.     

Synchronization for stochastic hybrid coupled controlled systems with Lévy noise

In this paper, synchronization for stochastic hybrid-delayed coupled systems with Lévy noise on a network (SHDCLN) is investigated via aperiodically intermittent control. Here time delays, Markovian switching and Lévy noise are considered on a network simultaneously for the first time. After that, by means of Lyapunov method, graph theory, and some techniques of inequality, some sufficient conditions are derived to guarantee the synchronization for SHDCLN. In addition, the designed range of aperiodically intermittent controller parameters is shown. Meanwhile, the coupling strength and the perturbed intensity of noise have a great impact on the intensity of control. Then, we investigate synchronization for stochastic hybrid delayed Chua's circuits with Lévy noise on a network as a practical application of our theoretical results. Finally, a numerical example is given to illustrate the effectiveness of the theoretical results.     

Adaptive coupled synchronization among multi-Lorenz systems family

In this paper, the adaptive synchronization method of coupled system is proposed for multi-Lorenz systems family. This method can avoid estimating the value of coupling coefficient. Strict theoretical proofs are given. And we derived a sufficient condition of synchronization for a general unidirectional coupling ring network with N identical Lorenz systems. The network is coupled through the first state variable of each equation. In fact, the whole unidirectional coupling network will synchronize by adding only one adaptive feedback gain equation. Numerical simulations show the effectiveness of the methods.     

Chaos control and synchronization of a modified chaotic system

By replacing a quadratic nonlinear term in Lü system with a piecewise linear signum (PWL) function, a new simplified three-dimensional piecewise continuous autonomous system (a modified Lü system) is introduced. The qualitative properties of the modified Lü system are studied. Based on these properties, the feedback control law is applied to suppress chaos to one of the three equilibria. Several different synchronized methods, such as the active control, one way coupling by active control, and the adaptive active control are applied to achieve the state synchronization of two identical modified Lü systems. These results show that after the simplification, the modified Lü system can still keep the basic and typical nonlinear phenomena. Compared with the original Lü system, the modified Lü system has a lot of advantages, by which the modified Lü system can be more easily implemented by theoretical analysis, and more practicable made by secret communications.     

Chaos and chaos synchronization for a non-autonomous rotational machine systems

Chaos and chaos synchronization of the centrifugal flywheel governor system are studied in this paper. By mechanics analyzing, the dynamical equation of the centrifugal flywheel governor system is established. Because of the non-linear terms of the system, the system exhibits both regular and chaotic motions. The characteristic of chaotic attractors of the system is presented by the phase portraits and power spectra. The evolution from Hopf bifurcation to chaos is shown by the bifurcation diagrams and a series of Poincaré sections under different sets of system parameters, and the bifurcation diagrams are verified by the related Lyapunov exponent spectra. This letter addresses control for the chaos synchronization of feedback control laws in two coupled non-autonomous chaotic systems with three different coupling terms, which is demonstrated and verified by Lyapunov exponent spectra and phase portraits. Finally, numerical simulations are presented to show the effectiveness of the proposed chaos synchronization scheme.     

Robust synchronization of fractional-order complex dynamical networks with parametric uncertainties

The study investigates robust synchronization of fractional-order complex dynamical networks with parametric uncertainties. Based on the properties of the kronecker product and the stability of the fractional-order system, the robust synchronization criteria are derived by applying the nonlinear control. These criteria are in the form of linear matrix inequalities which can be readily solved by applying the LMI toolbox. The coupling configuration matrix is not necessary to be symmetric or irreducible, and the inner coupling matrix needs not to be symmetric, diagonal or positive definite. Two numerical examples are provided to demonstrate the validity of the presented synchronization scheme.     

Control and synchronization of Julia sets in coupled map lattice

In this paper, we achieve the control and synchronization of Julia sets in coupled map lattice using gradient control and optimal control respectively. The control of the Julia sets is accomplished by controlling the stable space of the fixed plane. Moreover, the synchronization of two different Julia sets is also accomplished by their trajectories synchronization. To verify the feasibility of these control methods, we consider the Julia sets, whose lattice length is two, as examples to achieve their control and synchronization using different methods respectively. The numerical simulations are also shown to illustrate the effectiveness of these control methods.     

Chaos synchronization of the fractional-order Chen’s system

In this paper, based on the stability theorem of linear fractional systems, a necessary condition is given to check the chaos synchronization of fractional systems with incommensurate order. Chaos synchronization is studied by utilizing the Pecora–Carroll (PC) method and the coupling method. The necessary condition can also be used as a tool to confirm results of a numerical simulation. Numerical simulation results show the effectiveness of the necessary condition.     

New identification and control methods of sine-function Julia sets

In this paper, we propose two new methods to realize drive-response system synchronization control and parameter identification for two kinds of sine-function Julia sets. By means of these two methods, the zero asymptotic sliding variables and the stability theory in difference equations are applied to control the fractal identification. Furthermore, the problem of synchronization control is solved in the case of a drive system with unknown parameters, where the unknown parameters of the drive system can be identified in the asymptotic synchronization process. The results of simulation examples demonstrate the effectiveness of the new methods.