Practical bipartite synchronization via pinning control on a network of nonlinear agents with antagonistic interactions

This paper studies the synchronization phenomena in a network of heterogeneous nonlinear systems over signed graph, which can be considered as the perturbation of a network of homogeneous nonlinear systems. Assume that the signed graph is structurally balanced, the nonlinear system satisfies one-sided Lipschitz condition, and a leader pins a subset of agents. Under some proper initial conditions of the leader, we derive some conditions under which bipartite synchronization error can be kept arbitrary small by choosing a proper pining scheme. This property is called practical pinning bipartite synchronization as an alternative synchronization notion for network of heterogeneous nonlinear systems over signed graph. Finally, we present a numerical example to illustrate the effectiveness of the obtained results.     

The complete synchronization condition in a network of piezoelectric micro-beams

This work deals with the dynamics of a network of piezoelectric micro-beams (a stack of disks). The complete synchronization condition for this class of chaotic nonlinear electromechanical system with nearest-neighbor diffusive coupling is studied. The nonlinearities within the devices studied here are in both the electrical and mechanical components. The investigation is made for the case of a large number of coupled discrete piezoelectric disks. The problem of chaos synchronization is described and converted into the analysis of the stability of the system via its differential equations. We show that the complete synchronization of N identical coupled nonlinear chaotic systems having shift invariant coupling schemes can be calculated from the synchronization of two of them. According to analytical, semi-analytical predictions and numerical calculations, the transition boundaries for chaos synchronization state in the coupled system are determined as a function of the increasing number of oscillators.

     

Adaptive nonsingular terminal sliding mode control for synchronization of identical Φ 6 oscillators

The main goal of this paper is to propose the adaptive nonsingular terminal sliding mode controllers for complete synchronization (CS) and anti-synchronization (AS) between two identical ?? ⁶ Van der Pol or Duffing oscillators with presentations of system uncertainties and external disturbances. Unlike directly eliminating the nonlinear items of synchronized error system for sliding mode control schemes in the literature, the proposed adaptive controllers can tackle the nonlinear dynamics without active cancellation. The controllers can be implemented without known bounds of system uncertainties and external disturbances. Meanwhile, the feedback gains are not determined in advance but updated by the adaptive rules. Sufficient conditions are given based on the Lyapunov stability theorem and numerical simulations are performed to verify the effectiveness of presented schemes. The results show that the chaotic synchronization can be achieved accurately by the chattering free control.     

Transitions from strongly to weakly-nonlinear dynamics in a class of exactly solvable oscillators and nonlinear beat phenomena

In this paper, a regular perturbation tool is suggested to bridge the gap between weakly and strongly nonlinear dynamics based on exactly solvable oscillators with trigonometric characteristics considered by Nesterov (Proc. Mosc. Inst. Power Eng. 357:68–70, 1978). It is shown that the corresponding action-angle variables linearize the original oscillators with no special functions involved. As a result, linear and strongly nonlinear areas of the dynamics are described within the same perturbation procedure. The developed tool is applied then to analyzing the nonlinear beat and energy localization phenomena in two linearly coupled Duffing oscillators. It is shown that the principal phase variable describing the beat phenomena is governed by the hardening Nesterov oscillator with some perturbation due to qubic nonlinearity and coupling between the oscillators. As a result, the above class of strongly nonlinear oscillators is given clear physical meaning, whereas a closed form analytical solution is obtained for nonlinear beat and localization dynamics. Based on this solution, necessary and sufficient conditions for onset of energy localization are obtained.     

Synchronization of chaotic system with uncertain variable parameters by linear coupling and pragmatical adaptive tracking

We study the synchronization of general chaotic systems which satisfy the Lipschitz condition only, with uncertain variable parameters by linear coupling and pragmatical adaptive tracking. The uncertain parameters of a system vary with time due to aging, environment, and disturbances. A?sufficient condition is given for the asymptotical stability of common zero solution of error dynamics and parameter update dynamics by the Ge?CYu?CChen pragmatical asymptotical stability theorem based on equal probability assumption. Numerical results are studied for a Lorenz system and a quantum cellular neural network oscillator to show the effectiveness of the proposed synchronization strategy.     

Maps of Convex Sets and Invariant Regions¶for Finite-Difference Systems¶of Conservation Laws

General results about maps of convex sets in ? ⁿ are proved. We outline their extensions to an infinite-dimensional context. Such extensions have applications in nonlinear analysis such as in the study of the invariance of convex sets under nonlinear maps. Here, we explore applications only in the finite-dimensional context. More specifically, we apply the general results to the problem of finding sufficient conditions for a region of the state space to be globally or locally invariant under finite-difference schemes applied to systems of conservation laws in several space variables. In particular, we establish a final characterization of the invariant regions under the Lax-Friedrichs scheme and also give sufficient conditions for the local invariance. Further, we give sufficient conditions for the global and local invariance of regions under flux-splitting finite-difference schemes. An example of the multi-dimensional Euler equations for non-isentropic gas dynamics is discussed.     

Effect of coupling,synchronization of chaos and stick-slip motion in two mutually coupled dynamical systems

In this work, we study the synchronization of two coupled chaotic oscillators. The uncoupled system corresponds to a mass attached to a nonlinear spring and driven by a rolling carpet. For identical oscillators, complete synchronization is analyzed using Lyapunov stability theory. This first analysis reveals that stability area of synchronization increases with the values of the coupling coefficient. Numerical simulations are shown to illustrate and validate stick-slip and chaos synchronization. Some cases of anti-synchronization are detected. Curiously, amplification of fixed point either regular or chaotic is observed in the area of anti-synchronization. Furthermore, phase synchronization is studied for nonidentical oscillators. It appears that for certain values of the coupling coefficient, coincidence of the phases is obtained, while the amplitudes remain uncorrelated. Contrarily to the case of complete synchronization, it does not exist a threshold of the coupling from which phase synchronization could appear. Besides, when we add the modified tuned mass damper on the structure, the behavior of the system can change including the appearance of synchronization, particularly in the region of fixed point. More precisely, complete synchronization is improved in the region of fixed point, while the damage of synchronization is observed when the velocity of the carpets is less than \(0.30\) .     

Exponential synchronization for second-order nonlinear systems in complex dynamical networks with time-varying inner coupling via distributed event-triggered transmission strategy

This paper focuses on the exponential synchronization problem of complex dynamical networks (CDNs) with time-varying inner coupling via distributed event-triggered transmission strategy. Information update is driven by properly defined event, which depends on the measurement error. Each node is described as a second-order nonlinear dynamic system and only exchanges information with its neighboring nodes over a directed network. Suppose that the network communication topology contains a directed spanning tree. A sufficient condition for achieving exponential synchronization of second-order nonlinear systems in CDNs with time-varying inner coupling is derived. Detailed theoretical analysis on exponential synchronization is performed by the virtues of algebraic graph theory, distributed event-triggered transmission strategy, matrix inequality and the special Lyapunov stability analysis method. Moreover, the Zeno behavior is excluded as well by the strictly positive sampling intervals based on the upper right-hand Dini derivative. It is noted that the amount of communication among network nodes and network congestion have been significantly reduced so as to avoid the waste of network resources. Finally, a simulation example is given to show the effectiveness of the proposed exponential synchronization criteria.     

Generalized synchronization of the fractional-order chaos in weighted complex dynamical networks with nonidentical nodes

A fractional-order weighted complex network consists of a number of nodes, which are the fractional-order chaotic systems, and weighted connections between the nodes. In this paper, we investigate generalized chaotic synchronization of the general fractional-order weighted complex dynamical networks with nonidentical nodes. The well-studied integer-order complex networks are the special cases of the fractional-order ones. Based on the stability theory of linear fraction-order systems, the nonlinear controllers are designed to make the fractional-order complex dynamical networks with distinct nodes asymptotically synchronize onto any smooth goal dynamics. Numerical simulations are provided to verify the theoretical results. It is worth noting that the synchronization effect sensitively depends on both the fractional order ?? and the feedback gain k _i . Moreover, generalized synchronization of the fractional-order weighted networks can still be achieved effectively with the existence of noise perturbation.     

Distributed coordinated tracking of multiple autonomous underwater vehicles

This paper considers the distributed coordinated tracking problem of multiple autonomous underwater vehicles with a time-varying reference trajectory. Each vehicle is subject to model uncertainty and time-varying ocean disturbances. A novel predictor-based neural dynamic surface control design approach is proposed to develop the node controllers, under which synchronization between vehicles can be reached on condition that the augmented graph induced by the vehicles and the reference trajectory contains a spanning tree. The prediction errors are used to update the neural adaptive laws, which enable fast identifying the vehicle dynamics without excessive knowledge of their dynamical models. Further, this result is extended to the output-feedback case where only position-yaw information can be measured. A local predictor, based on its own position-yaw information, is constructed, not only to recover the unmeasured velocity information, but also to identify the unknown dynamics for each vehicle. A linear matrix inequality-based analysis is performed for the stability of the predictor. Then, distributed output-feedback tracking controllers are developed to achieve synchronization between vehicles in the presence of unknown dynamics and unmeasured velocities. For both cases, the stability properties of the closed-loop network are established via Lyapunov analysis. Simulation results demonstrate the effectiveness of the proposed methods.     

Synchronization of improved chaotic Colpitts oscillators using nonlinear feedback control

The model and the normalized state equations of the novel version of the Colpitts oscillator designed to operate in the ultra-high frequency range are presented. The circuit is investigated numerically and simulations demonstrate chaos in the microwave frequency range. Typical phase portrait, Lyapunov exponent and Lyapunov dimension are calculated using a piece-wise linear approximation of nonlinear I–V characteristic of the bipolar junction transistor. In addition, the feedback controller is applied to achieve chaos synchronization for two identical improved chaotic Colpitts oscillators. In the frame the nonlinear function of the system is used as a nonlinear feedback term for the stability of the error dynamics. Finally, numerical simulations show that this control method is feasible for this oscillator.     

Adaptive projective synchronization of dynamical networks with distributed time delays

In this paper, the adaptive projective synchronization of dynamical network with distributed time delays is investigated. Network with unknown topology and network with both unknown topology and system parameters of node dynamics are considered respectively. Based on Lyapunov stability theory and LaSalle’s invariance principle, the sufficient conditions for achieving projective synchronization are obtained. Numerical examples are provided to show the effectiveness of the proposed method.     

Adaptive feedback control and synchronization of non-identical chaotic fractional order systems

This paper addresses the reliable synchronization problem between two non-identical chaotic fractional order systems. In this work, we present an adaptive feedback control scheme for the synchronization of two coupled chaotic fractional order systems with different fractional orders. Based on the stability results of linear fractional order systems and Laplace transform theory, using the master-slave synchronization scheme, sufficient conditions for chaos synchronization are derived. The designed controller ensures that fractional order chaotic oscillators that have non-identical fractional orders can be synchronized with suitable feedback controller applied to the response system. Numerical simulations are performed to assess the performance of the proposed adaptive controller in synchronizing chaotic systems.     

Intelligent quadratic optimal synchronization of?uncertain chaotic systems via LMI approach

This paper proposes an intelligent quadratic optimal control scheme via linear matrix inequality (LMI) approach for the synchronization of uncertain chaotic systems with both external disturbances and parametric perturbations. First, a four-layered neural fuzzy network (NFN) identifier is constructed to estimate system nonlinear dynamics. Based on the NFN identifier, an intelligent quadratic optimal controller is developed with robust hybrid control scheme, in which H _?? optimal control and variable structure control (VSC) are embedded to attenuate the effects of external disturbances and parametric perturbations. The adaptive tuning laws of network parameters are derived in the sense of the Lyapunov synthesis approach to ensure network convergence, and the sufficient criterion for existence of the controller is formulated in the linear matrix inequality (LMI) form to guarantee the quadratic optimal synchronization performance. Finally, a numerical simulation example is illustrated by the chaotic Chua??s circuit system to demonstrate the effectiveness of our scheme.     

Chaos synchronization of two different chaotic complex Chen and Lü systems

This paper investigates the phenomenon of chaos synchronization of two different chaotic complex systems of the Chen and Lü type via the methods of active control and global synchronization. In this regard, it generalizes earlier work on the synchronization of two identical oscillators in cases where the drive and response systems are different, the parameter space is larger, and the dimensionality increases due to the complexification of the dependent variables. The idea of chaos synchronization is to use the output of the drive system to control the response system so that the output of the response system converges to the output of the drive system as time increases. Lyapunov functions are derived to prove that the differences in the dynamics of the two systems converge to zero exponentially fast, explicit expressions are given for the control functions and numerical simulations are presented to illustrate the success of our chaos synchronization techniques. We also point out that the global synchronization method is better suited for synchronizing identical chaotic oscillators, as it has serious limitations when applied to the case where the drive and response systems are different.     

Synchronization in the Lorenz system: Stability and robustness

We present necessary and sufficient conditions for the existence of synchronization in a class of continuous-time nonlinear systems: the so-called -affine systems. We apply the results to the Lorenz attractor. The robustness of the synchronization against parameter value variations is discussed using the Lyapunov stability theory for perturbed systems. We obtain sufficient conditions that guarantee a bounded steady-state error. This technique gives conservative results; however, in some systems like that of Lorenz, it provides definitive results about the existence of the synchronization. Furthermore, we give estimates of the maximal error as a function of the difference between the parameter values of the systems to be synchronized.     

The synchronization of general complex dynamical network via pinning control

In this paper, the globally synchronization of the general complex network is investigated. Firstly, we discuss the synchronization problem of the linearly coupled and directed network under the pinning control, and make comparison with the previous work about the undirected network. Sufficient conditions are obtained to guarantee the realization of synchronization. Secondly, the synchronization problem of nonlinearly coupled and undirected network under the pinning control is studied, and a criteria of getting synchronization is given. Furthermore, we introduced the adaptive adjustment of the coupling strength in nonlinearly coupled network. At last, we give simulation examples to verify our theoretical results.     

Observer-based synchronization in fractional-order leader–follower complex networks

This paper is devoted to synchronization behavior of complex dynamical networks with a Caputo fractional-order derivative. In particular, we propose a fractional-order leader–follower complex network where the leader is independent, has its own dynamics, and is followed by all the other nodes. Using the state observer approach, the leader and the followers are designed to connect through only a scalar coupling signal instead of the commonly used full state coupling scheme. On the basis of stability theory of the fractional-order differential system, a sufficient condition for global network synchronization is presented, which only relates to the linear part of individual nodes and can be easily solved by the pole-placement design technique. The analytic results are complemented with numerical simulations for a network whose nodes are governed by the fractional-order Chua’s circuit.     

Synchronization and chaos control by quorum sensing mechanism

Diverse rhythms are generated by thousands of oscillators that somehow manage to operate synchronously. By using mathematical and computational modeling, we consider the synchronization and chaos control among chaotic oscillators coupled indirectly but through a quorum sensing mechanism. Some sufficient criteria for synchronization under quorum sensing are given based on traditional Lyapunov function method. The Melnikov function method is used to theoretically explain how to suppress chaotic Lorenz systems to different types of periodic oscillators in quorum sensing mechanics. Numerical studies for classical Lorenz and Rössler systems illustrate the theoretical results.     

Anticipatory, complete and lag synchronization of chaos and hyperchaos in a nonlinear delay-coupled time-delayed system

We study the synchronization of chaos and hyperchaos in first-order time-delayed systems that are coupled using the nonlinear time-delay excitatory coupling. We assign two characteristic time delays: the system delay that is same for both the systems, and the coupling delay associated with the coupling path. We show that depending upon the relative values of the system delay and the coupling delay the coupled systems show anticipatory, complete, and lag synchronization. We derive a general stability condition for all the synchronization processes using the Krasovskii–Lyapunov theory. Numerical simulations are carried out to corroborate the analytical results. We compute a quantitative measure to ensure the occurrence of different synchronization phenomena. Finally, we set up an experiment in electronic circuit to verify all the synchronization scenario. It is observed that the experimental results are in good agreement with our analytical results and numerical observations.