Synchronization of different fractional order chaotic systems using active control

Synchronization of fractional order chaotic dynamical systems is receiving increasing attention owing to its interesting applications in secure communications of analog and digital signals and cryptographic systems. In this article we utilize active control technique to synchronize different fractional order chaotic dynamical systems. Further we investigate the interrelationship between the (fractional) order and synchronization in different chaotic dynamical systems. It is observed that synchronization is faster as the order tends to one.     

Synchronization of fractional order chaotic systems using active control method

In this article, the active control method is used for synchronization of two different pairs of fractional order systems with Lotka–Volterra chaotic system as the master system and the other two fractional order chaotic systems, viz., Newton–Leipnik and Lorenz systems as slave systems separately. The fractional derivative is described in Caputo sense. Numerical simulation results which are carried out using Adams–Bashforth–Moulton method show that the method is easy to implement and reliable for synchronizing the two nonlinear fractional order chaotic systems while it also allows both the systems to remain in chaotic states. A salient feature of this analysis is the revelation that the time for synchronization increases when the system-pair approaches the integer order from fractional order for Lotka–Volterra and Newton–Leipnik systems while it reduces for the other concerned pair.     

Adaptive fuzzy sliding mode control for synchronization of uncertain fractional order chaotic systems

This paper deals with chaos synchronization between two different uncertain fractional order chaotic systems based on adaptive fuzzy sliding mode control (AFSMC). With the definition of fractional derivatives and integrals, a fuzzy Lyapunov synthesis approach is proposed to tune free parameters of the adaptive fuzzy controller on line by output feedback control law and adaptive law. Moreover, chattering phenomena in the control efforts can be reduced. The sliding mode design procedure not only guarantees the stability and robustness of the proposed AFSMC, but also the external disturbance on the synchronization error can be attenuated. The simulation example is included to confirm validity and synchronization performance of the advocated design methodology.     

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.     

Modified projective synchronization of chaotic systems with disturbances via active sliding mode control

We apply the active sliding mode control technique to realize the modified projective synchronization of the chaotic systems. The disturbances are considered both in the drive system and the response system. The sufficient conditions for the modified projective synchronization both the non-identical and identical chaotic systems are presented. The corresponding numerical simulations are provided to illuminate the effectiveness of the proposed active sliding mode controllers.     

Stabilization and synchronization of chaotic systems via intermittent control

In this paper, we consider the stabilization and synchronization of chaotic systems via intermittent control with time varying control period and control width. Compared to existing results, some less conservative conditions are derived to guarantee the stabilization of nonlinear system. An effective adaptive-intermittent control law is also presented. Two examples are given to verify our proposed results.     

Reduced-order synchronization of uncertain chaotic systems via adaptive control

We consider the coupling of two uncertain dynamical systems with different orders using an adaptive feedback linearization controller to achieve reduced-order synchronization between the two systems. Reduced-order synchronization is the problem of synchronization of a slave system with projection of a master system. The synchronization scheme is an exponential linearizing-like controller and a state/uncertainty estimator. As an illustrative example, we show that the dynamical evolution of a second-order driven oscillator can be synchronized with the canonical projection of a fourth-order chaotic system. Simulation results indicated that the proposed control scheme can significantly improve the synchronousness performance. These promising results justify the usefulness of the proposed output feedback controller in the application of secure communication.     

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.     

Global chaos synchronization of new chaotic systems via nonlinear control

Nonlinear control is an effective method for making two identical chaotic systems or two different chaotic systems be synchronized. However, this method assumes that the Lyapunov function of error dynamic (e) of synchronization is always formed as V (e) = 1/2e^Te. In this paper, modification based on Lyapunov stability theory to design a controller is proposed in order to overcome this limitation. The method has been applied successfully to make two identical new systems and two different chaotic systems (new system and Lorenz system) globally asymptotically synchronized. Since the Lyapunov exponents are not required for the calculation, this method is effective and convenient to synchronize two identical systems and two different chaotic systems. Numerical simulations are also given to validate the proposed synchronization approach.     

Stabilizing periodic orbits of fractional order chaotic systems via linear feedback theory

In this paper stabilizing unstable periodic orbits (UPO) in a chaotic fractional order system is studied. Firstly, a technique for finding unstable periodic orbits in chaotic fractional order systems is stated. Then by applying this technique to the fractional van der Pol and fractional Duffing systems as two demonstrative examples, their unstable periodic orbits are found. After that, a method is presented for stabilization of the discovered UPOs based on the theories of stability of linear integer order and fractional order systems. Finally, based on the proposed idea a linear feedback controller is applied to the systems and simulations are done for demonstration of controller performance.     

On control and synchronization in chaotic and hyperchaotic systems via linear feedback control

This paper presents the control and synchronization of chaos by designing linear feedback controllers. The linear feedback control problem for nonlinear systems has been formulated under optimal control theory viewpoint. Asymptotic stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function which can clearly be seen to be the solution of the Hamilton–Jacobi–Bellman equation thus guaranteeing both stability and optimality. The formulated theorem expresses explicitly the form of minimized functional and gives the sufficient conditions that allow using the linear feedback control for nonlinear system. The numerical simulations were provided in order to show the effectiveness of this method for the control of the chaotic Rössler system and synchronization of the hyperchaotic Rössler system.     

Synchronization of the unified chaotic systems via active control

This paper investigates the synchronization of coupled unified chaotic systems via active control. The synchronization is given in the slave–master scheme and the controller ensures that the states of the controlled chaotic slave system exponentially synchronize with the state of the master system. Numerical simulations are provided for illustration and verification of the proposed method.     

Image encryption based on synchronization of fractional chaotic systems

This paper deals with a synchronization scheme for two fractional chaotic systems which is applied in image encryption. Based on Pecora and Carroll (PC) synchronization, fractional-order Lorenz-like system forms a master–slave configuration, and the sufficient conditions are derived to realize synchronization between these two systems via the Laplace transformation theory. An image encryption algorithm is introduced where the original image is encoded by a nonlinear function of a fractional chaotic state. Simulation results show that the original image is well masked in the cipher texts and recovered successfully through chaotic signals. Further, the cryptanalysis is conducted in detail through histogram, information entropy, key space and sensitivity to verify the high security.     

Chaos synchronization between two different chaotic systems via nonlinear feedback control

This work presents chaos synchronization between two different chaotic systems via nonlinear feedback control. On the basis of a converse Lyapunov theorem and balanced gain scheme, control gains of controller are derived to achieve chaos synchronization for the unified chaotic systems. Numerical simulations are shown to verify the results.     

Controlling general projective synchronization of fractional order Rossler systems

This paper proposed a method to achieve general projective synchronization of two fractional order Rossler systems. First, we construct the fractional order Rossler system’s corresponding approximation integer order system. Then, a control method based on a partially linear decomposition and negative feedback of state errors was utilized on the integer order system. Numerical simulations show the effectiveness of the proposed method.     

Modified projective synchronization of uncertain fractional order hyperchaotic systems

Base on the stability theory of fractional order system, this work mainly investigates modified projective synchronization of two fractional order hyperchaotic systems with unknown parameters. A controller is designed for synchronization of two different fractional order hyperchaotic systems. The method is successfully applied to modified projective synchronization between fractional order Rössler hyperchaotic system and fractional order Chen hyperchaotic system, and numerical simulations illustrate the effectiveness of the obtained results.     

Adaptive synchronization of new fractional‐order chaotic systems with fractional adaption laws based on risk analysis

In this paper, a new fractional‐order chaotic system and an adaptive synchronization of fractional‐order chaotic system are proposed. Parameters adaption laws are obtained to design adaptive controllers using Lyapunov stability theory of fractional‐order system. Finally, reliability of designed controllers and risk analysis of adaptive synchronization problem are formulated and, risk of using the proposed controllers in presences of external disturbances are demonstrated. Also, risk of controllers are reduced using an optimizing method. Numerical examples are used to verify the performance of the proposed controllers.     

Robust stabilization and synchronization of a class of fractional-order chaotic systems via a novel fractional sliding mode controller

This paper proposes a novel fractional-order sliding mode approach for stabilization and synchronization of a class of fractional-order chaotic systems. Based on the fractional calculus a stable integral type fractional-order sliding surface is introduced. Using the fractional Lyapunov stability theorem, a single sliding mode control law is proposed to ensure the existence of the sliding motion in finite time. The proposed control scheme is applied to stabilize/synchronize a class of fractional-order chaotic systems in the presence of model uncertainties and external disturbances. Some numerical simulations are performed to confirm the theoretical results of the paper. It is worth noticing that the proposed fractional-order sliding mode controller can be applied to control a broad range of fractional-order dynamical systems.     

Adaptive synchronization of uncertain chaotic systems via backstepping design

In this paper, an approach for adaptive synchronization of uncertain chaotic systems is proposed using adaptive backstepping with tuning functions. Strong properties of global stability and asymptotic synchronization can be achieved. The proposed approach offers a systematic design procedure for adaptive synchronization of a large class of continuous-time chaotic systems in the chaos research literature. Simulation results are presented to show the effectiveness of the approach.     

Robust synchronization of drive–response chaotic systems via adaptive sliding mode control

A robust adaptive sliding control scheme is developed in this study to achieve synchronization for two identical chaotic systems in the presence of uncertain system parameters, external disturbances and nonlinear control inputs. An adaptation algorithm is given based on the Lyapunov stability theory. Using this adaptation technique to estimate the upper-bounds of parameter variation and external disturbance uncertainties, an adaptive sliding mode controller is then constructed without requiring the bounds of parameter and disturbance uncertainties to be known in advance. It is proven that the proposed adaptive sliding mode controller can maintain the existence of sliding mode in finite time in uncertain chaotic systems. Finally, numerical simulations are presented to show the effectiveness of the proposed control scheme.