Synchronization of nonidentical chaotic fractional-order systems with different orders of fractional derivatives

This paper addresses the problem of synchronization of chaotic fractional-order systems with different orders of fractional derivatives. Based on the stability theory of fractional-order linear systems and the idea of tracking control, suitable controllers are correspondingly proposed for two cases: the first is synchronization between two identical chaotic fractional-order systems with different fractional orders, and the other is synchronization between two nonidentical fractional-order chaotic systems with different fractional orders. Three numerical examples illustrate that fast synchronization can be achieved even between a chaotic fractional-order system and a hyperchaotic fractional-order system.     

Projective synchronization between two different time-delayed chaotic systems using active control approach

In this paper, we investigate the projective synchronization between two different time-delayed chaotic systems. A suitable controller is chosen using the active control approach. We relax some limitations of previous work, where projective synchronization of different chaotic systems can be achieved only in finite dimensional chaotic systems, so we can achieve projective synchronization of different chaotic systems in infinite dimensional chaotic systems. Based on the Lyapunov stability theory, we suggest a generic method to achieve the projective synchronization between two different time-delayed chaotic systems. The validity of the proposed method is demonstrated and verified by observing the projective synchronization between two well-known time-delayed chaotic systems; the Ikeda system and Mackey–Glass system. Numerical simulations fully support the analytical approach.     

Finite-time combination-combination synchronization of four different chaotic systems with unknown parameters via sliding mode control

In this paper, we apply the nonsingular terminal sliding mode control technique to realize the novel combination-combination synchronization between combination of two chaotic systems as drive system and combination of two chaotic systems as response system with unknown parameters in a finite time. On the basic of the adaptive laws and finite-time stability theory, an adaptive combination sliding mode controller is proposed to ensure the occurrence of the sliding motion in a given finite time for four different chaotic systems. In theory, it is proved that the sliding mode technique can realize fast convergence for four different chaotic systems in the finite time. Some criteria and corollaries are derived for finite-time combination-combination synchronization of four different chaotic systems. Numerical simulation results are shown to verify the effectiveness and correctness of the combination-combination synchronization.     

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.     

Synchronization of N different coupled chaotic systems with ring and chain connections

Synchronization of N different coupled chaotic systems with ring and chain Lorenz system, and the R(o)ssler system are used as examples in verifying effectiveness of the method. Based on the Lyapunov stability theory, the form of the controller is designed and the area of the coupling coefficients is determined. Simulations indicate that global synchronization of the N different chaotic systems can be realized by choosing appropriate coupling coefficients by using the controller.     

Crack synchronization of chaotic circuits under field coupling

Nonlinear electric devices are important and essential for setting circuits so that chaotic outputs or periodical series can be generated. Chaotic circuits can be mapped into dimensionless dynamical systems by using scale transformation, and thus, synchronization control can be further investigated in numerical way. In case of synchronization approach, resistor is often used to bridge two chaotic circuits and gap junction connection is used to realize possible synchronization. In fact, complex electromagnetic effect in circuits should be considered when the capacitor and inductor (inductance coil) are attacked by high-frequency signals or noise-like disturbance. In this paper, two chaotic circuits are connected by using voltage coupling (via resistor) and triggering mutual induction electromotive force, which time-varying magnetic field is generated in the inductance coils. Therefore, magnetic field coupling is realized between two isolate inductance coils and induction electromotive force is generated to adjust the oscillation in circuits. It is found that field coupling can modulate the synchronization behaviors of chaotic circuits. In case of periodical oscillating state, the synchronization between two periodical circuits under voltage coupling is destroyed when field coupling is considered. Furthermore, the synchronization between chaotic circuits becomes more difficult when field coupling is triggered. Open problems for this topic are proposed for further investigation.     

Lag synchronization between discrete chaotic systems with diverse structure

A lag synchronization controller is designed in studying discrete chaotic systems with diverse structures to realize synchronization between Henon and Ikeda sys- terns. The structure of the lag synchronization controller and the error equations of state variables between discrete chaotic systems are presented based on the stability theory. The designed controller has unique structures for different chaotic systems. Lag synchro- nization between any discrete chaotic systems with diverse structures can be achieved. Simulation results show that this control method is effective and feasible.     

Finite-time generalized synchronization of chaotic systems with different order

In this paper, the generalized synchronization of chaotic systems with different order is studied. The definition of finite-time generalized synchronization is put forward for the first time. Based on the finite-time stability theory, two control strategies are proposed to realize the generalized synchronization of chaotic systems with different order in finite time. Besides the relation between the parameter β, the initial states of systems and the convergent time were obtained. The corresponding numerical simulations are presented to demonstrate the effectiveness of proposed schemes.     

Generalized finite-time synchronization between coupled chaotic systems of different orders with unknown parameters

This letter investigates the adaptive finite-time synchronization of different coupled chaotic (or hyperchaotic) systems with unknown parameters. The sufficient conditions for achieving the generalized finite-time synchronization of two chaotic systems are derived based on the theory of finite-time stability of dynamical systems. By the adaptive control technique, the control laws and the corresponding parameters update laws are proposed such that the generalized finite-time synchronization of nonidentical chaotic (or hyperchaotic) systems is to be obtained. These results obtained are in good agreement with the existing one in open literature and it is shown that the technique introduced here can be further applied to various finite-time synchronizations between dynamical systems. Finally, numerical simulations are given to demonstrate the effectiveness of the proposed scheme.     

Complete (anti-)synchronization of chaotic systems with fully uncertain parameters by adaptive control

This paper addresses a unified mathematical expression describing a class of chaotic systems, for which the problem of synchronization and anti-synchronization between different chaotic systems with fully uncertain parameters and different structure are studied. Based on the Lyapunov stability theory, a novel, simple, and systemic adaptive synchronization controller is designated, the analytic expression of the controller and the adaptive laws of parameters are developed. Moreover, the proposed scheme can be extended to anti-synchronize a class of chaotic systems. Two chaotic systems with different structure and fully uncertain parameters are employed as the examples to show the effectiveness of the proposed adaptive synchronization and anti-synchronization schemes. Additionally, the robustness and noise immunity of the adaptive synchronization scheme is investigated by measuring the mean squared error of the systems.     

Hybrid phase synchronization between identical and nonidentical three-dimensional chaotic systems using the active control method

In this article, the active control method is used to investigate the hybrid phase synchronization between two identical Rikitake and Windmi systems, and also between two nonidentical systems taking Rikitake as the driving system and Windmi system as the response system. Based on the Lyapunov stability theory, the sufficient conditions for achieving the hybrid phase synchronization of two chaotic systems are derived. The active control method is found to be very effective and convenient to achieve hybrid phase chaos synchronization of the identical and nonidentical chaotic systems. Numerical simulation results which are carried out using the Runge–Kutta method show its feasibility and effectiveness for the synchronization of dynamical chaotic systems.     

Synchronization of N different coupled chaotic systems with ring and chain connections

Synchronization of N different coupled chaotic systems with ring and chain connections is investigated. The New system, the Chen system, the Lii system, the Lorenz system, and the Rossler system are used as examples in verifying effectiveness of the method. Based on the Lyapunov stability theory, the form of the controller is designed and the area of the coupling coefficients is determined. Simulations indicate that global synchronization of the N different chaotic systems can be realized by choosing appropriate coupling coefficients by using the controller.     

Dynamics analysis and hybrid function projective synchronization of a new chaotic system

This paper introduces a novel three-dimensional autonomous chaotic system by adding a quadratic cross-product term to the first equation and modifying the state variable in the third equation of a chaotic system proposed by Cai et al. (Acta Phys. Sin. 56:6230, 2007). By means of theoretical analysis and computer simulations, some basic dynamical properties, such as Lyapunov exponent spectrum, bifurcations, equilibria, and chaotic dynamical behaviors of the new chaotic system are investigated. Furthermore, hybrid function projective synchronization (HFPS) of the new chaotic system is studied by employing three different synchronization methods, i.e., adaptive control, system coupling and active control. The proposed approaches are applied to achieve HFPS between two identical new chaotic systems with fully uncertain parameters, HFPS in coupled new chaotic systems, and HFPS between the integer-order new chaotic system and the fractional-order Lü chaotic system, respectively. Corresponding numerical simulations are provided to validate and illustrate the analytical results.     

Synchronization between integer-order chaotic systems and a class of fractional-order chaotic system based on fuzzy sliding mode control

In this paper, we focus on the synchronization between integer-order chaotic systems and a class of fractional-order chaotic system using the stability theory of fractional-order systems. A new fuzzy sliding mode method is proposed to accomplish this end for different initial conditions and number of dimensions. Furthermore, three examples are presented to illustrate the effectiveness of the proposed scheme, which are the synchronization between a fractional-order Lü chaotic system and an integer-order Liu chaotic system, the synchronization between a fractional-order hyperchaotic system based on Chen??s system and an integer-order hyperchaotic system based upon the Lorenz system, and the synchronization between a fractional-order hyperchaotic system based on Chen??s system, and an integer-order Liu chaotic system. Finally, numerical results are presented and are in agreement with theoretical analysis.     

Parameter-dependent synchronization of coupled neurons in cold receptor model

In this paper, synchronization in two coupled neurons with spiking, bursting and chaos firings is investigated as the coupling strength gets increased. Synchronization state can be identified by means of the bifurcation diagram, the correlation coefficient and ISI-distance. It is illustrated that the coupled neurons can exhibit different types of synchronization state when the coupling strength increases. The different synchronization processes appear similar, but their detailed processes are different depending on the parameter values. The synchronization of neuronal network with two different network connectivity patterns is also studied. It is shown that chaotic and high period pattern are more difficult to get complete synchronization than the situation in single spike and low period pattern. It is also demonstrated that the synchronization status of multiple neurons is dependent on the network connectivity patterns. These results may be instructive to understand synchronization in neuronal systems.     

Generalized synchronization of discrete systems

Generalized synchronization of two discrete systems was discussed. By constructing appropriately nonlinear coupling terms, some sufficient conditions for determining the generalized synchronization between the drive and response systems were derived. In a positive invariant and bounded set, many chaotic maps satisfy the sufficient conditions. The effectiveness of the sufficient conditions is illustrated by three examples.     

Phase synchronization between nonlinearly coupled Rössler systems

Phase synchronization between nonlinearly coupled systems with 1：1 and 1：2 resonances is investigated. By introducing a concept of phase for a chaotic motion, it is demonstrated that for different internal resonances, with relatively small parameter epsilon, the difference between the mean frequencies of the two sub-oscillators approaches zero. This implies that phase synchronization can be achieved for weak interaction between the two oscillators. With the increase in coupling strength, fluctuations of the frequency difference can be observed, and for the primary resonance, the amplitudes of the fluctuations of the difference seem much smaller compared to the case with frequency ratio 1：2, even with the weak coupling strength. Unlike the enhanced effect on synchronization for linear coupling, the increase in nonlinear coupling strength results in the transition from phase synchronization to a non-synchronized state. Further investigation reveals that the states from phase synchronization to non-synchronization are related to the critical changes of the Lyapunov exponents, which can also be explained with the diffuse clouds.     

Nonlinear active observer-based generalized synchronization in time-delayed systems

When two different chaotic oscillators are coupled, generalized synchronization can occur. It may imply a very complicated relation between the states of drive and response systems. We propose a method that can be used to detect and characterize the generalized synchronization in modulated time-delayed systems. Using Krasovskii–Lyapunov theory, sufficient condition for generalized synchronization is derived. The proposed technique has been applied to synchronize prototype and Ikeda models by numerical simulation.     

Q–S synchronization between chaotic systems with double scaling functions

A double function Q–S synchronization (DFQSS) scheme of non-identical chaotic systems is proposed and analyzed with the assumption that all of the parameters are unknown. The sufficient conditions for achieving the double function Q–S synchronization with the desired scaling functions of two different chaotic systems (including the systems of non-identical dimension) are derived based on Lyapunov stability theory. By the adaptive control technique, the control laws and the corresponding parameter update laws are presented such that the DFQSS of non-identical chaotic systems is to be achieved. Numerical simulations and a brief discussion conclude the paper.     

Active coupling and its circuitry designs of chaotic systems against deteriorated and delayed networks

This paper is concerned with the problem of asymptotic synchronization of a class of chaotic systems in the presence of network deterioration and time-varying delays. Based on adaptive adjustment technique and circuitry principle, a new version of the active coupling as well as its circuit realization is proposed. Then, an approach that is based on application of Lyapunov stability theory for the synchronization error system is introduced to prove the asymptotic synchronization result of the overall chaotic system. Moreover, a condition which denotes that at least one coupling will not be deteriorated for synchronization of the network is provided in the paper. It is shown that, without control inputs, the result can also be established for the deteriorated coupling networks and any time-varying bounded delay under the topological structure satisfying the condition. Finally, the proposed active couplings are physically implemented by circuits and tested by simulation on a Chua??s circuit network.