Synchronizing different chaotic systems using active sliding mode control

An active sliding mode controller is designed to synchronize three pairs of different chaotic systems (Lorenz–Chen, Chen–Lü, and Lü–Lorenz) in drive–response structure. It is assumed that the system parameters are known. The closed loop error dynamics depend on the linear part of the response systems and parameters of the controller. Therefore, the synchronization rate can be adjusted through these parameters. Analysis of the stability for the proposed method is derived based on the Lyapunov stability theorem. Finally, numerical results are presented to show the effectiveness of the proposed control technique.     

Finite-time synchronization of uncertain unified chaotic systems based on CLF

This paper studies the problem of finite-time synchronization for the unified chaotic systems. We prove that global finite-time synchronization can be achieved for unified chaotic systems which have uncertain parameters. Simulation results for Lorenz, Lü and Chen chaotic systems are provided to illustrate the effectiveness of the proposed scheme.     

Adaptive anti-synchronization of chaotic complex nonlinear systems with unknown parameters

This paper presents the adaptive anti-synchronization of a class of chaotic complex nonlinear systems described by a united mathematical expression with fully uncertain parameters. Based on Lyapunov stability theory, an adaptive control scheme and adaptive laws of parameters are developed to anti-synchronize two chaotic complex systems. The anti-synchronization of two identical complex Lorenz systems and two different complex Chen and Lü systems are taken as two examples to verify the feasibility and effectiveness of the presented scheme.     

Adaptive reduced-order anti-synchronization of chaotic systems with fully unknown parameters

In this paper, we investigate the reduced-order anti-synchronization of uncertain chaotic systems. Based upon the parameters modulation and the adaptive control techniques, we show that dynamical evolution of third-order chaotic system can be anti-synchronized with the canonical projection of a fourth-order chaotic system even though their parameters are unknown. The techniques are successfully applied to two examples: hyperchaotic Lorenz system (fourth-order) and Lorenz system (third-order); Lü hyperchaotic system (fourth-order) and Chen system (third-order). Theoretical analysis and numerical simulations are shown to verify the results.     

Synchronization of unified chaotic systems with uncertain parameters based on the CLF

Chaos synchronization in unified chaotic systems with uncertain parameters is discussed in this paper. On the basis of the control Lyapunov function (CLF), a feedback controller is designed which is only related to the boundaries of the uncertain parameters. Synchronization of two identical unified chaotic systems with different initial conditions is realized. Simulation results for Lorenz, Lü and Chen chaotic systems are provided to verify the effectiveness of the proposed scheme.     

Finite-time chaos synchronization of unified chaotic system with uncertain parameters

This paper deals with the finite-time chaos synchronization of the unified chaotic system with uncertain parameters. Based on the finite-time stability theory, a control law is proposed to realize finite-time chaos synchronization for the unified chaotic system with uncertain parameters. The controller is simple, robust and only part parameters are required to be bounded. Simulation results for the Lorenz, Lü and Chen chaotic systems are presented to validate the design and the analysis.     

Chaos synchronization between two different chaotic systems using active control

This work presents chaos synchronization between two different chaotic systems by using active control. This technique is applied to achieve chaos synchronization for each pair of the dynamical systems Lorenz, Lü and Chen. Numerical simulations are shown to verify the results.     

A new scheme of general hybrid projective complete dislocated synchronization

Based on the Lyapunov stability theorem, a new type of chaos synchronization, general hybrid projective complete dislocated synchronization (GHPCDS), is proposed under the framework of drive-response systems. The difference between the GHPCDS and complete synchronization is that every state variable of drive system does not equal the corresponding state variable, but equal other ones of response system while evolving in time. The GHPCDS includes complete dislocated synchronization, dislocated anti-synchronization and projective dislocated synchronization as its special item. As examples, the Lorenz chaotic system, Rössler chaotic system, hyperchaotic Chen system and hyperchaotic Lü system are discussed. Numerical simulations are given to show the effectiveness of these methods.     

Chaos control of chaotic dynamical systems using backstepping design

This work presents chaos control of chaotic dynamical systems by using backstepping design method. This technique is applied to achieve chaos control for each of the dynamical systems Lorenz, Chen and Lü systems. Based on Lyapunov stability theory, control laws are derived. We used the same technique to enable stabilization of chaotic motion to a steady state as well as tracking of any desired trajectory to be achieved in a systematic way. Numerical simulations are shown to verify the results.     

Adaptive fuzzy control for a class of chaotic systems with nonaffine inputs

A robust adaptive fuzzy control scheme is presented for a class of chaotic systems with nonaffine inputs, modeling uncertainties and external disturbances by using backstepping approach. Fuzzy logic systems (FLS) are employed to approximate the unknown parts of the virtual control and practical controls. The main characteristics of the scheme are that the number of the online adaptive parameters is no more than two times of the order of chaotic system and the tracking errors are guaranteed to be uniformly asymptotically stable with the aid of additional adaptive compensation terms. Lorenz system, Chen system, Lü system and Liu system are presented to illustrate the feasibility and effectiveness of the proposed control technique.     

Occurrence and underlying mechanism of multi-stripe chaotic attractors

This paper is concerned with the generation of multi-stripe chaotic attractors. Simple periodic nonlinear functions are employed to transform the original chaotic attractors to a pattern with multiple “parallel” or “rectangular” stripes. The relationship between the system parameters related to some periodic functions and the shape of the generated attractor is analyzed. Theoretic analysis about the underlying mechanism of generating the parallel stripes in the attractors is given. A general creation mechanism of multi-stripe attractors of the Lorenz system and other well-known chaotic systems is derived from the proposed unified approach.     

Simplified method and synchronization for a class of complex chaotic systems

This paper is devoted to investigate a class of complex chaotic systems and a linear correlation between the real and imaginary component of complex variables in these systems is found. Based on this linear relationship, a simplified law is proposed. First, complex Lorenz system is given to show the linear correlation, then it is simplified. Second, a simplified law is proposed to determine whether the complex system can be simplified, and the complex Lü system and hyperchaotic complex Lü system are used to verify the simplified law. Finally, a new synchronization control is proposed to synchronize complex Lorenz system and real Lorenz system. The theoretical analysis and numerical simulation prove the feasibility and better performance of this method.     

Bifurcation analysis of some forced Lu systems

Bifurcation behaviour of a forced Lu system is analyzed as the system parameter c and a forcing parameter F are varied. The Lu system belongs to a family of generalized Lorenz system. Members of this family are known to exhibit different types of chaotic attractors. Some of these attractors have been named Lorenz type L, Lu or Transition type T, Chen type T and Transverse 8 Type S. These different types of chaotic attractors are visually distinct when the parameters are widely separated. However, there is a need for identifying the precise point where transition from one type of chaotic attractor to another takes place. We identified signatures in the return map, which could be used for determining the point of transition and classifying the different types of chaotic attractors. These signatures helped to identify the point in coordinate space associated with such transitions. We find that such transitions take place when a chaotic attractor comes very close to a one-dimensional manifold on which the time derivatives of two of the variables is zero. We also find that just before coming to this point in coordinate space associated with the transition, the trajectory had approached, very closely, the equilibrium point at the origin.     

When Darwin meets Lorenz: Evolving new chaotic attractors through genetic programming

In this paper, we propose a novel methodology for automatically finding new chaotic attractors through a computational intelligence technique known as multi-gene genetic programming (MGGP). We apply this technique to the case of the Lorenz attractor and evolve several new chaotic attractors based on the basic Lorenz template. The MGGP algorithm automatically finds new nonlinear expressions for the different state variables starting from the original Lorenz system. The Lyapunov exponents of each of the attractors are calculated numerically based on the time series of the state variables using time delay embedding techniques. The MGGP algorithm tries to search the functional space of the attractors by aiming to maximise the largest Lyapunov exponent (LLE) of the evolved attractors. To demonstrate the potential of the proposed methodology, we report over one hundred new chaotic attractor structures along with their parameters, which are evolved from just the Lorenz system alone.     

Projective synchronization between different fractional‐order hyperchaotic systems with uncertain parameters using proposed modified adaptive projective synchronization technique

In the present article, the authors have proposed a modified projective adaptive synchronization technique for fractional‐order chaotic systems. The adaptive projective synchronization controller and identification parameters law are developed on the basis of Lyapunov direct stability theory. The proposed method is successfully applied for the projective synchronization between fractional‐order hyperchaotic Lü system as drive system and fractional‐order hyperchaotic Lorenz chaotic system as response system. A comparison between the effects on synchronization time due to the presence of fractional‐order time derivatives for modified projective synchronization method and proposed modified adaptive projective synchronization technique is the key feature of the present article. Numerical simulation results, which are carried out using Adams–Boshforth–Moulton method show that the proposed technique is effective, convenient and also faster for projective synchronization of fractional‐order nonlinear dynamical systems. Copyright © 2013 John Wiley & Sons, Ltd.     

Full state hybrid projective synchronization in continuous-time chaotic (hyperchaotic) systems

Chaos synchronization, as an important topic, has become an active research subject in nonlinear science. Over the past two decades, chaos synchronization between nonlinear systems has been extensively studied, and many types of synchronization have been announced. This paper introduces another novel type of chaos synchronization – full state hybrid projective synchronization (FSHPS), which includes complete synchronization, anti-synchronization and projective synchronization as its special item. Based on the Lyapunov’s direct method, the general FSHPS scheme is given and illustrated with Lorenz chaotic system and hyperchaotic Chen system as examples. Numerical simulations are used to verify the effectiveness of the proposed scheme.     

A stable-manifold-based method for chaos control and synchronization

A stable-manifold-based method is proposed for chaos control and synchronization. The novelty of this new and effective method lies in that, once the suitable stable manifold according to the desired dynamic properties is constructed, the goal of control is only to force the system state to lie on the selected stable manifold because once the stable manifold is reached, the chaotic system will be guided towards the desired target. The effectiveness of the approach and idea is tested by stabilizing the Newton–Leipnik chaotic system which possesses more than one strange attractor and by synchronizing the unified chaotic system which unifies both the Lorenz system and the Chen system.     

A universal unfolding of the Lorenz system

A universal unfolding of the Lorenz system is derived and studied in this paper. Both rigorous theoretical analysis and numerical simulations show that the Lorenz system, the Chen system, and the Lü system belong to the same universal unfolding. Therefore, they all have similar dynamical behaviors in the sense that if the Lorenz system has limit cycles produced from a Hopf bifurcation for a certain set of parameter values, then the other two systems also have limit cycles from the same set of parameter values; and if the Lorenz, Chen, and Lü systems are chaotic for some parameter values (for example, some typical parameter values), respectively, then the homotopic system for the Lorenz system and the Chen system, and the homotopic system for these three systems, are all chaotic within the entire domain of these homotopic parameters.     

Parameter identification and synchronization of fractional-order chaotic systems

The knowledge about parameters and order is very important for synchronization of fractional-order chaotic systems. In this article, identification of parameters and order of fractional-order chaotic systems is converted to an optimization problem. Particle swarm optimization algorithm is used to solve this optimization problem. Based on the above parameter identification, synchronization of the fractional-order Lorenz, Chen and a novel system (commensurate or incommensurate order) is derived using active control method. The new fractional-order chaotic system has four-scroll chaotic attractors. The existence and uniqueness of solutions for the new fractional-order system are also investigated theoretically. Simulation results signify the performance of the work.     

Synchronizing chaotic systems using control based on a special matrix structure and extending to fractional chaotic systems

This work presents a direct approach to design stabilizing controller based on a special matrix structure to synchronize chaotic systems and extends the approach to synchronize fractional chaotic systems. With this method, chaos synchronization is implemented in Lorenz chaotic systems with known parameters and the same to Lorenz chaotic systems with unknown parameters. Especially, fractional Lorenz chaotic system with unknown parameters is synchronized by fractional Chen chaotic system too. Numerical simulations confirm the effectiveness of the method proposed.