Synchronized states in a ring of four mutually coupled self-sustained electromechanical devices

In this paper, we study the dynamics of a ring of four mutually coupled identical self-sustained electromechanical devices both in their autonomous and nonautonomous chaotic states. The transition boundaries that can occur between instability and complete synchronization states when the coupling strength varies are derived. Numerical simulations are then performed to support the accuracy of the analytical approach.     

Synchronized states in a ring of four mutually coupled oscillators and experimental application to secure communications

Data encryption has become increasingly important for many applications including phone, internet and satellite communications. Considering the desirable properties of ergodicity and high sensitivity to initial conditions and control parameters, chaotic signals are suitable for encryption systems. Chaotic encryption systems generally have high speed with low cost, which makes them better candidates than many traditional ciphers for multimedia data encryption. In this paper, analytical and numerical methods as well as experimental implementation are used to prove partial and complete synchronized states in a ring of four autonomous oscillators in their chaotic states. Application to secure communication is discussed.     

Cluster states in a ring of four coupled semiconductor lasers

By varying the coupling strength, clusters synchronization is found in a ring of four mutually coupled semiconductor lasers with optical feedback. The numerical simulation confirms the mathematical analysis and shows the degree of synchronization in terms of the parameters differences between the lasers.     

Synchronization of coupled chaotic FitzHugh-Nagumo systems

This paper addresses dynamic synchronization of two FitzHugh-Nagumo (FHN) systems coupled with gap junctions. All the states of the coupled chaotic system, treating either as single-input or two-input control system, are synchronized by stabilizing their error dynamics, using simplest and locally robust control laws. The local asymptotic stability, chosen by utilizing the local Lipschitz nonlinear property of the model to address additionally the non-failure of the achieved synchronization, is ensured by formulating the matrix inequalities on the basis of Lyapunov stability theory. In the presence of disturbances, it ensures the local uniform ultimate boundedness. Furthermore, the robustness of the proposed methods is ensured against bounded disturbances besides providing the upper bound on disturbances. To the best of our knowledge, this is the computationally simplest solution for synchronization of coupled FHN modeled systems along with unique advantages of less conservative local asymptotic stability of synchronization errors with robustness. Numerical simulations are carried out to successfully validate the proposed control strategies.     

Synchronization of multiple bursting neurons ring coupled via impulsive variables

Synchronization behavior of bursting neurons is investigated in a neuronal network ring impulsively coupled, in which each neuron exhibits chaotic bursting behavior. Based on the Lyapunov stability theory and impulsive control theory, sufficient conditions for synchronization of the multiple systems coupled with impulsive variables can be obtained. The neurons become synchronous via suitable impulsive strength and resetting period. Furthermore, the result is obtained that synchronization among neurons is weakened with the increasing of the reset period and the number of neurons. Finally, numerical simulations are provided to show the effectiveness of the theoretical results.© 2014 Wiley Periodicals, Inc. Complexity 21: 29–37, 2015     

Synchronization dynamics of two different dynamical systems

In this paper, synchronization dynamics of two different dynamical systems is investigated through the theory of discontinuous dynamical systems. The necessary and sufficient conditions for the synchronization, de-synchronization and instantaneous synchronization (penetration or grazing) are presented. Using such a synchronization theory, the synchronization of a controlled pendulum with the Duffing oscillator is systematically discussed as a sampled problem, and the corresponding analytical conditions for the synchronization are presented. The synchronization parameter study is carried out for a better understanding of synchronization characteristics of the controlled pendulum and the Duffing oscillator. Finally, the partial and full synchronizations of the controlled pendulum with periodic and chaotic motions are presented to illustrate the analytical conditions. The synchronization of the Duffing oscillator and pendulum are investigated in order to show the usefulness and efficiency of the methodology in this paper. The synchronization invariant domain is obtained. The technique presented in this paper should have a wide spectrum of applications in engineering. For example, this technique can be applied to the maneuvering target tracking, and the others.     

Synchronization and antisynchronization of N‐coupled fractional‐order complex chaotic systems with ring connection

This paper is devoted to investigate synchronization and antisynchronization of N‐coupled general fractional‐order complex chaotic systems described by a unified mathematical expression with ring connection. By means of the direct design method, the appropriate controllers are designed to transform the fractional‐order error dynamical system into a nonlinear system with antisymmetric structure. Thus, by using the recently established result for the Caputo fractional derivative of a quadratic function and a fractional‐order extension of the Lyapunov direct method, several stability criteria are derived to ensure the occurrence of synchronization and antisynchronization among N‐coupled fractional‐order complex chaotic systems. Moreover, numerical simulations are performed to illustrate the effectiveness of the proposed design.     

Synchronization of the coupled FitzHugh–Nagumo systems

Buri and Todorovi recently studied the coupling of FitzHugh–Nagumo equations that can be used to describe neurological signal propagation. Using techniques from modern control theory, a nonlinear controller is obtained that results in two of the Buri and Todorovi systems being synchronized with each other. The detailed derivation that leads to the controller and numerical results that verify the controller’s ability to synchronize the two systems together are included.     

Synchronization of N-coupled incommensurate fractional-order chaotic systems with ring connection

In this paper, based on the stability theorem of linear fractional systems, a necessary condition is given to synchronize N-coupled incommensurate fractional-order chaotic systems with ring connection. Chaos synchronization is studied by utilizing the coupling method that includes unidirectional and bidirectional coupling. The main contribution of this work is to design the coupling parameters in which it is shown that our proposed controller does stabilize error dynamics around all of equilibriums of master system. Finally, the claims are justified through a set of simulations and results coincide with the theoretical analysis.     

Cooperative dynamics of strongly coupled distributed systems

Doklady Mathematics -     

Synchronization of N-coupled fractional-order chaotic systems with ring connection

In this paper, the synchronization of N-coupled fractional-order chaotic systems with ring connection is firstly investigated in detail. Based on stability criteria of fractional-order system, the synchronization of N-coupled fractional-order chaotic systems with unidirectional coupling and bidirectional coupling is achieved. Moreover, some appropriate comparisons are made to contrast to some of existing results. Finally, some numerical examples are provided to illustrate and verify the effectiveness of the proposed schemes.     

Synchronization of delayed coupled reaction‐diffusion systems on networks

In this paper, the exponential synchronization problem of delayed coupled reaction‐diffusion systems on networks (DCRDSNs) is investigated. Based on graph theory, a systematic method is designed to achieve exponential synchronization between two DCRDSNs by constructing a global Lyapunov function for error system. Two different kinds of sufficient synchronization criteria are derived in the form of Lyapunov functions and coefficients of drive‐response systems, respectively. Finally, a numerical example is given to show the usefulness of the proposed criteria. Copyright © 2014 John Wiley & Sons, Ltd.     

Global synchronization of three coupled chaotic systems with ring connection

This paper studies chaos synchronization of three coupled chaos systems with ring connection. New generic criteria of global chaos synchronization are proposed respectively according to the way of coupling (unidirectional or bidirectional). As an example, The criteria are successfully applied to three coupled identical Lorenz systems. Numerical simulation are shown for demonstration.     

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 coupled systems to periodic diagonal solutions with synchronized asymptotic phases

Using the change of coordinates, parameterization and characteristic multipliers, we prove the synchronization of a class of coupled nonlinear systems with nontrivial periodic solution. The periodic diagonal solution of the coupled system is asymptotically orbitally stable with asymptotic phase.Examples are given to illustrate the theorem.     

Synchronization of two coupled systems of J-J type using active sliding mode control

This paper first presents a rigorous mathematical analysis for the active sliding mode control method proposed recently by Haeri and Emadzadeh. Second, the technique is applied to achieve synchronization between two coupled systems of J-J. Numerical simulations are used to verify the above analytical results.     

Synchronization of stochastic coupled systems via feedback control based on discrete-time state observations

In this paper, we formulate and investigate the synchronization of stochastic coupled systems via feedback control based on discrete-time state observations (SCSFD). The discrete-time state feedback control is used in the drift parts of response system. Combining Lyapunov method with graph theory, the upper bound of duration between two consecutive state observations is provided. And a global Lyapunov function of SCSFD is presented, which derives some sufficient criteria to guarantee the synchronization of drive–response systems in the sense of mean-square asymptotical synchronization. In addition, the theoretical results are applied to stochastic coupled oscillators and second-order Kuramoto oscillators. Finally, two numerical examples are given to verify the effectiveness of the theoretical results.     

Noise-induced on–off intermittency in mutually coupled semiconductor lasers

Intermittent switches between low-frequency fluctuations and steady-state emission are experimentally observed in two bidirectionally coupled semiconductor lasers subject to common Gaussian noise applied to the laser pump currents. The time series analysis reveals power-law scalings typical for on–off intermittency near its onset, with critical exponents of −1 for the mean turbulent length versus noise intensity and −3/2 for probability distribution of laminar phases versus the laminar length. The same −1 power-law scaling is found by the power spectrum analysis for the signal-to-noise ratio versus the noise intensity.     

Synchronization for the coupled stochastic strict-feedback nonlinear systems with delays under pinning control

In this paper, a class of new coupled stochastic strict-feedback nonlinear systems with delays (CSFND) on networks without strong connectedness (NWSC) is considered, and the issue pertaining to the synchronization of the systems is discussed by pinning control. Towards CSFND, the controllers are approached by combining the back-stepping method and the design of virtual controllers. A key novel design ingredient is that the global Lyapunov function is obtained based on each Lyapunov function of stochastic strict-feedback nonlinear systems with delays (SFND). Moreover, a sufficient criterion is presented to realize the exponential synchronization by employing the graph theory and Lyapunov method. As a subsequent result, we apply the obtained theoretical results to the second-order oscillator systems and robotic arm systems. Meanwhile, numerical simulations are provided to demonstrate the validity and feasibility of our theoretical results.