Nonlinear dynamics and synchronization of coupled electromechanical systems with multiple functions

This paper deals with the nonlinear dynamics and synchronization of coupled electromechanical systems with multiple functions, described by an electrical Duffing oscillator magnetically coupled to linear mechanical oscillators. Firstly, the amplitudes of the sub- and super-harmonic oscillations for the resonant states are obtained and discussed using the multiple time scales method. The equations of motion are solved numerically using the Runge–Kutta algorithm. It is found that chaotic and periodic orbit coexist in the electromechanical system depending on the set of initial conditions. Secondly, the problem of synchronization dynamics of two coupled electromechanical systems both in the regular and chaotic states is also investigated, and estimation of the coupling coefficient under which synchronization and no-synchronization take place is made.     

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.     

Vibration analysis and bifurcations in the self-sustained electromechanical system with multiple functions

We consider in this paper the dynamics of the self-sustained electromechanical system with multiple functions, consisting of an electrical Rayleigh–Duffing oscillator, magnetically coupled with linear mechanical oscillators. The averaging and the harmonic balance method are used to find the amplitudes of the oscillatory states respectively in the autonomous and nonautonomous cases, and analyze the condition in which the quenching of self-sustained oscillations appears. The influence of system parameters as well as the number of linear mechanical oscillators on the bifurcations in the response of this electromechanical system is investigated. Various bifurcation structures, the stability chart and the variation of the Lyapunov exponent are obtained, using numerical simulations of the equations of motion.     

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.     

One- and two-cluster synchronized dynamics of non-diffusively coupled Tchebycheff map networks

We use the master stability formalism to discuss one- and two-cluster synchronization of coupled Tchebycheff map networks. For diffusively coupled map systems, the one-cluster synchronized dynamics is given by the behaviour of the individual maps, and the coupling only determines the stability of the coherent state. For the case of non-diffusive coupling and for two-cluster synchronization, the synchronized dynamics on networks is different from the behaviour of the single individual map. Depending on the coupling, we study numerically the characteristics of various forms of the resulting synchronized dynamics. The stability properties of the respective one-cluster synchronized states are discussed for arbitrary network structures. For the case of two-cluster synchronization on bipartite networks we also present analytical expressions for fixed points and zig-zag patterns, and explicitly determine the linear stability of these orbits for the special case of ring-networks.     

Synchronization dynamics in a ring of four mutually coupled biological systems

This paper considers the synchronization dynamics in a ring of four mutually coupled biological systems described by coupled Van der Pol oscillators. The coupling parameter are non-identical between oscillators. The stability boundaries of the process are first evaluated without the influence of the local injection using the eigenvalues properties and the fourth-order Runge–Kutta algorithm. The effects of a locally injected trajectory on the stability boundaries of the synchronized states are performed using numerical simulations. In both cases, the stability boundaries and the main dynamical states are reported on the stability maps in the (K₁, K₂) plane.     

Generalized synchronization of identical chaotic systems on the route from an independent dynamics to the complete synchrony

The transition from asynchronous hyperchaos to complete synchrony in coupled identical chaotic systems may either occur directly or be mediated by a preliminary stage of generalized synchronization. In the present paper we investigate the underlying mechanisms of realization of the both scenarios. It is shown that a generalized synchronization arises when the manifold of identically synchronous states M is transversally unstable, while the local transversal contraction of phase volume first appears in the areas of phase space separated from M and being visited by the chaotic trajectories. On the other hand, a direct transition from an asynchronous hyperchaos to the complete synchronization occurs, under variation of the controlling parameter, if the transversal stability appears first on the manifold M, and only then it extends upon the neighboring phase volume. The realization of one or another scenario depends upon the choice of the coupling function. This result is valid for both unidirectionally and mutually coupled systems, that is confirmed by theoretical analysis of the discrete models and numerical simulations of the physically realistic flow systems.     

Mutual synchronization behavior for chaotic systems via limited capacity communication channels

Several important properties of chaos synchronization of bidirectional coupled systems remain still unexplored. This article investigates synchronization behavior for chaotic systems subject to states quantization. Based on the invariance principle of differential equations, an adaptive feedback scheme is proposed to strictly synchronize chaotic systems via limited capacity communication channels. Furthermore, it is important to point out that the mutual synchronization behavior for bidirectional coupled systems is determined by the amount of transmitting information and the initial states of coupled systems. © 2015 Wiley Periodicals, Inc. Complexity 21: 335–342, 2016     

Strange attractors and synchronization dynamics of coupled Van der Pol–Duffing oscillators

We consider in this paper the synchronization dynamics of coupled chaotic Van der Pol–Duffing systems. We first find that with the judicious choose of the set of initial conditions, the model exhibits two strange chaotic attractors. The problem of synchronizing chaos both on the same and different chaotic orbits of two coupled Van der Pol–Duffing systems is investigated. The stability boundaries of the synchronization process between two coupled driven Van der Pol model are derived and the effects of the amplitude of the periodic perturbation of the coupling parameter on these boundaries are analyzed. The results are provided on the stability map in the (q, K) plane.     

On analytical justification of phase synchronization in different chaotic systems

In analytical or numerical synchronizations studies of coupled chaotic systems the phase synchronizations have less considered in the leading literatures. This article is an attempt to find a sufficient analytical condition for stability of phase synchronization in some coupled chaotic systems. The method of nonlinear feedback function and the scheme of matrix measure have been used to justify this analytical stability, and tested numerically for the existence of the phase synchronization in some coupled chaotic systems.     

Universal chaos synchronization control laws for general quadratic discrete systems

This paper addresses the reliable universal synchronization problem between two coupled chaotic quadratic discrete systems. A general nonlinear control method of synchronization for coupled 2D and 3D quadratic dynamical systems in discrete-time is proposed. The proposed synchronization method is based on universal controllers. The synchronization results are derived theoretically using active control method and Lyapunov stability theory. Numerical simulations are performed to assess the performance of the presented analysis.     

Weak synchronization of chaotic neural networks with parameter mismatch via periodically intermittent control

This paper deals with the synchronization of two coupled identical chaotic systems with parameter mismatch via using periodically intermittent control. In general, parameter mismatches are considered to have a detrimental effect on the synchronization quality between coupled identical systems: in the case of small parameter mismatches the synchronization error does not decay to zero or even a nonzero mean. Larger values of parameter mismatches can even result in the loss of synchronization. via intermittent control with periodically intervals, we can obtain the weak synchronization. Some sufficient conditions for the stabilization and weak synchronization of a large class of coupled identical chaotic systems will be derived by using Lyapunov stability theory. The analytical results are confirmed by numerical simulations.     

Numerical stability of chaotic synchronization using a nonlinear coupling function

Nonlinear coupling has been used to synchronize some chaotic systems. The difference evolutional equation between coupled systems, determined via the linear approximation, can be used to analyze the stability of the synchronization between drive and response systems. According to the stability criteria the coupled chaotic systems are asymptotically synchronized, if all eigenvalues of the matrix found in this linear approximation have negative real parts. There is no synchronization, if at least one of these eigenvalues has positive real part. Nevertheless, in this paper we have considered some cases on which there is at least one zero eigenvalue for the matrix in the linear approximation. Such cases demonstrate synchronization-like behavior between coupled chaotic systems if all other eigenvalues have negative real parts.     

Adaptive synchronization between two different chaotic dynamical systems

This work presents the synchronization between two different chaotic systems by using an adaptive feedback control scheme. The adaptive synchronization problem between an electrostatic system and electromechanical transducer has been investigated. An adaptive linear feedback law with two controllers is proposed to ensure the global chaos synchronization of the nonlinear electrostatic and electromechanical systems. Numerical simulations results are presented to demonstrate the effectiveness of the proposed method.     

A note on phase synchronization in coupled chaotic fractional order systems

The dynamic behaviors of fractional order systems have received increasing attention in recent years. This paper addresses the reliable phase synchronization problem between two coupled chaotic fractional order systems. An active nonlinear feedback control scheme is constructed to achieve phase synchronization between two coupled chaotic fractional order systems. We investigated the necessary conditions for fractional order Lorenz, Lü and Rössler systems to exhibit chaotic attractor similar to their integer order counterpart. Then, based on the stability results of fractional order systems, sufficient conditions for phase synchronization of the fractional models of Lorenz, Lü and Rössler systems are derived. The synchronization scheme that is simple and global enables synchronization of fractional order chaotic systems to be achieved without the computation of the conditional Lyapunov exponents. Numerical simulations are performed to assess the performance of the presented analysis.     

A New Approach to Synchronization Analysis of Linearly Coupled Map Lattices

In this paper, a new approach to analyze synchronization of linearly coupled map lattices (LCMLs) is presented. A reference vector x(t) is introduced as the projection of the trajectory of the coupled system on the synchronization manifold. The stability analysis of the synchronization manifold can be regarded as investigating the difference between the trajectory and the projection. By this method, some criteria are given for both local and global synchronization. These criteria indicate that the left and right eigenvectors corresponding to the eigenvalue "0" of the coupling matrix play key roles in the stability of synchronization manifold for the coupled system. Moreover, it is revealed that the stability of synchronization manifold for the coupled system is different from the stability for dynamical system in usual sense. That is, the solution of the coupled system does not converge to a certain knowable s(t) satisfying s(t 1) = f(s(t)) but to the reference vector on the synchronization manifold, which in fact is a certain weighted average of each xi(t) for i = 1, ... ,m, but not a solution s(t) satisfying s(t 1) = f(s(t)).     

Exponential generalized synchronization of uncertain coupled chaotic systems by adaptive control

In this paper, the exponential generalized synchronization for a class of coupled systems with uncertainties is defined. A novel and powerful method is proposed to investigate the generalized synchronization based on the adaptive control technique. According to the Lyapunov stability theory, rigorous proof is given for the exponential stability of error system. In comparison with previous schemes, the presented method shortens the synchronization time and is more applicable in practice. Besides, it is shown that the synchronization effect is robust against the uncertain factors. Some typical chaotic and hyper-chaotic systems are taken as examples to illustrate above approach. The corresponding numerical simulations are demonstrated to verify the effectiveness of proposed method.     

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.     

Finite‐time synchronization of a class of N‐coupled complex partial differential systems with time‐varying delay

This study examines finite‐time synchronization for a class of N‐coupled complex partial differential systems (PDSs) with time‐varying delay. The problem of finite‐time synchronization for coupled drive‐response PDSs with time‐varying delay is similarly considered. The synchronization error dynamic of the PDSs is defined in the q‐dimensional spatial domain. We construct a feedback controller to achieve finite‐time synchronization. Sufficient conditions are derived by using the Lyapunov‐Krasoviskii stability approach and inequalities technology to ensure that the proposed networks achieve synchronization in finite time. The proposed systems demonstrate extensive application. Finally, an example is used to verify the theoretical results.     

Stability criterion for synchronization of linearly coupled unified chaotic systems

This paper investigates the synchronization of two linearly coupled unified chaotic systems. A new stability criterion for asymptotic synchronization is attained using the Lyapunov stability theory and linear matrix inequality (LMI) approach. A numerical example is given to illuminate the design procedure and advantage of the result derived.