The generalized Q-S synchronization between the generalized Lorenz canonical form and the Rössler system

In this paper, we investigate the generalized Q-S synchronization between the generalized Lorenz canonical form and the Rössler system. Firstly, we transform an arbitrary generalized Lorenz system to the generalized Lorenz canonical form, and the relation between the parameter of the generalized Lorenz system and the parameter of the generalized Lorenz canonical form are shown. Secondly, we extend the scheme present by [Yan ZY. Chaos 2005;15:023902] to study the generalized Q-S synchronization between the generalized Lorenz canonical form and the Rössler system, the more general controller is obtained. By choosing different parameter in the generalized controller obtained here, without much extra effort, we can get the controller of synchronization between the Chen system and the Rössler system, the Lü system and the Rössler system, the classic Lorenz system and the Rössler system, the Hyperbolic Lorenz system and the Rössler system, respectively. Finally, numerical simulations are used to perform such synchronization and verify the effectiveness of the controller.     

Chaos in a generalized Lorenz system

A three-component dynamic system describing a quantum cavity electrodynamic device with a pumping and nonlinear dissipation is studied. Various dynamical regimes are investigated in terms of divergent trajectories approaches and fractal statistics. It has been shown that stable and unstable dissipative structures type of limit cycles can be formed in such system, with variation of pumping and nonlinear dissipation rates. Transitions to chaotic regime and the corresponding chaotic attractor are studied in detail.     

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.     

Coexistence of anti-phase and complete synchronization in the generalized Lorenz system

This paper investigates a class of new synchronization phenomenon. Some control strategy is established to guarantee the coexistence of anti-phase and complete synchronization in the generalized Lorenz system. The efficiency of the control scheme is revealed by some illustrative simulations.     

Chaos control of new Ikeda–Lorenz systems by GYC partial region stability theory

A new strategy to achieve chaos control by GYC partial region stability theory is proposed. By using the GYC partial region stability theory, the Lyapunov function is a simple linear homogeneous function of error states, the controllers are more simple and have less simulation error because they are in lower degree than that of traditional controllers. Simulation results for a new Ikeda–Lorenz system show the effectiveness of this strategy. Copyright © 2008 John Wiley & Sons, Ltd.     

Master–slave synchronization of Lorenz systems using a single controller

A single controller for synchronization of two Lorenz systems is obtained by using Lyapunov function. Numerical results are given for the all three cases with one controller in each equation. Controller contains two or three variables of the master system.     

Optimal control and synchronization of Lorenz system with complete unknown parameters

The paper discusses the optimal control and synchronization problems of Lorenz systems with fully unknown parameters. Based on the Liapunov–Bellman technique, the optimal control law with three-state variables feedback is derived such that the trajectory of the Lorenz system is optimally stabilized to an equilibrium point of the uncontrolled system. Further, another optimal control law is also applied to achieve the state synchronization of two identical Lorenz systems. Numerical results to demonstrate the effectiveness of the proposed control scheme.     

Adaptive synchronization for two identical generalized Lorenz chaotic systems via a single controller

This paper presents a systematic design procedure to synchronize two identical generalized Lorenz chaotic systems based on a sliding mode control. In contrast to the previous works, this approach only needs a single controller to realize synchronization, which has considerable significance in reducing the cost and complexity for controller implementation. A switching surface only including partial states is adopted to ensure the stability of the error dynamics in the sliding mode. Then an adaptive sliding mode controller (ASMC) is derived to guarantee the occurrence of the sliding motion even when the parameters of the drive and response generalized Lorenz systems are unknown. Last, an example is included to illustrate the results developed in this paper.     

Chaos synchronization of a fractional‐order modified Van der Pol–Duffing system via new linear control,backstepping control and Takagi–Sugeno fuzzy approaches

In this article, based on the stability theory of fractional‐order systems, chaos synchronization is achieved in the fractional‐order modified Van der Pol–Duffing system via a new linear control approach. A fractional backstepping controller is also designed to achieve chaos synchronization in the proposed system. Takagi‐Sugeno fuzzy models‐based are also presented to achieve chaos synchronization in the fractional‐order modified Van der Pol–Duffing system via linear control technique. Numerical simulations are used to verify the effectiveness of the synchronization schemes. © 2015 Wiley Periodicals, Inc. Complexity 21: 116–124, 2016     

Chaos synchronization of a new hyperchaotic system

In this paper, two kinds of synchronization schemes for a new hyperchaotic system are presented. Firstly, on the basis of stability criterion of linear system, synchronization is achieved with the help of the active control theory. Secondly, a nonlinear controller is designed according to Lyapunov stability theory to assure that synchronization can be achieved. Furthermore, an adaptive control approach for synchronization of uncertain hyperchaotic systems is proposed. Finally numerical simulations are provided to show the effectiveness and feasibility of the developed methods.     

Impulsive control and synchronization of the Lorenz systems family

In this paper, impulsive control and synchronization for the newly presented Lorenz systems family are systematically investigated. Some new and more comprehensive criteria for global exponential stability and asymptotical stability of impulsively controlled Lorenz systems family are established with varying impulsive intervals. In particular, several simple and easily verified criteria are derived with equivalent impulsive intervals. An illustrative example is also provided to show the effectiveness and feasibility of the impulsive control method.     

Chaos synchronization between single and double wells Duffing–Van der Pol oscillators using active control

This paper presents chaos synchronization between single and double wells Duffing–Van der Pol (DVP) oscillators with Φ⁴ potential based on the active control technique. The technique is applied to achieve global synchronization between identical double-well DVP oscillators, identical single-well DVP oscillators and non-identical DVP oscillators, consisting of the double-well and the single-well DVP oscillators, respectively. Numerical simulations are also presented to verify the analytical results.     

Chaos synchronization and chaos control of quantum-CNN chaotic system by variable structure control and impulse control

In this paper, we derive some less stringent conditions for the exponential and asymptotic stability of impulsive control systems with impulses at fixed times. These conditions are then used to design an impulsive control law for the Quantum Cellular Neural Network chaotic system, which drives the chaotic state to zero equilibrium and synchronizes two chaotic systems. An active sliding mode control method is synchronizing two chaotic systems and controlling chaotic state to periodic motion state. And a sufficient condition is drawn for the robust stability of the error dynamics, and is applied to guiding the design of the controllers. Finally, numerical results are used to show the robustness and effectiveness of the proposed control strategy.     

Chaos synchronization of an energy resource system based on linear control

Synchronization of an energy resource system is investigated. Three linear control schemes are proposed to synchronize a chaotic energy resource system via the back-stepping method. This can be viewed as an improvement to the existing results of Tian et al. (2006) [14]. Because we use simpler controllers to realize a global asymptotical synchronization. In the first two schemes, the sufficient conditions for achieving synchronization of two identical energy resource systems using linear feedback control are derived by using Lyapunov stability theorem. In the third scheme, the synchronization condition is obtained by numerical method, in which only one state variable controller is contained. Finally, three numerical simulation examples are performed to verify these results.     

New estimate the bounds for the generalized Lorenz system

The bound of a chaotic system is important for chaos control, chaos synchronization, and other applications. In the present paper, the bounds of the generalized Lorenz system are studied, based on the Lyapunov function theory and the Lagrange multiplier method. We obtain a precise bound for the generalized Lorenz system. The rate of the trajectories is also obtained. Furthermore, we perform the numerical simulations. Numerical simulations are presented to show the effectiveness of the proposed scheme. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Dynamical behavior of a generalized Lorenz system model and its simulation

In this article, the dynamical behavior of a generalized Lorenz system is derived based on stability theory of dynamical systems. The meaningful contribution of this article is that the domain of attraction of the new chaotic system is studied in detailed. Finally, numerical simulations are given to verify the effectiveness and correctness of the obtained results. © 2015 Wiley Periodicals, Inc. Complexity 21: 99–105, 2016     

Chaos in fractional conjugate Lorenz system and its scaling attractors

Chaotic dynamics of fractional conjugate Lorenz system are demonstrated in terms of local stability and largest Lyapunov exponent. Chaos does exist in the fractional conjugate Lorenz system with order less than three since it has positive largest Lyapunov exponent. Furthermore, scaling chaotic attractors of fractional conjugate Lorenz system is theoretically and numerically analyzed with the help of one-way synchronization method and adaptive synchronization method. Numerical simulations are performed to verify the theoretical analysis.     

Chaos in fractional-order Genesio-Tesi system and its synchronization

In this paper we study the chaotic dynamics of fractional-order Genesio-Tesi system. Theoretically, a necessary condition for occurrence of chaos is obtained. Numerical investigations on the dynamics of this system have been carried out and properties of the system have been analyzed by means of Lyapunov exponents. It is shown that in case of commensurate system the lowest order of fractional-order Genesio-Tesi system to yield chaos is 2.79. Further, chaos synchronization of fractional-order Genesio-Tesi system is investigated via two different control strategies. Active control and sliding mode control are proposed and the stability of the controllers are studied. Numerical simulations have been carried out to verify the effectiveness of controllers.     

Chaos control and generalized projective synchronization of heavy symmetric chaotic gyroscope systems via Gaussian radial basis adaptive variable structure control

This paper proposes the chaos control and the generalized projective synchronization methods for heavy symmetric gyroscope systems via Gaussian radial basis adaptive variable structure control. Because of the nonlinear terms of the gyroscope system, the system exhibits chaotic motions. Occasionally, the extreme sensitivity to initial states in a system operating in chaotic mode can be very destructive to the system because of unpredictable behavior. In order to improve the performance of a dynamic system or avoid the chaotic phenomena, it is necessary to control a chaotic system with a periodic motion beneficial for working with a particular condition. As chaotic signals are usually broadband and noise like, synchronized chaotic systems can be used as cipher generators for secure communication. This paper presents chaos synchronization of two identical chaotic motions of symmetric gyroscopes. In this paper, the switching surfaces are adopted to ensure the stability of the error dynamics in variable structure control. Using the neural variable structure control technique, control laws are established which guarantees the chaos control and the generalized projective synchronization of unknown gyroscope systems. In the neural variable structure control, Gaussian radial basis functions are utilized to on-line estimate the system dynamic functions. Also, the adaptation laws of the on-line estimator are derived in the sense of Lyapunov function. Thus, the unknown gyro systems can be guaranteed to be asymptotically stable. Also, the proposed method can achieve the control objectives. Numerical simulations are presented to verify the proposed control and synchronization methods. Finally, the effectiveness of the proposed methods is discussed.