Controlling hyperchaos in hyperchaotic Lorenz system using feedback controllers

In this paper, the control strategies for hyperchaotic Lorenz system is investigated. The ordinary, dislocated, enhancing and speed feedback controls are used to suppress hyperchaos to unstable equilibrium. The Routh-Hurwitz criterion is used to derive the conditions of stability of the controlled hyperchaotic systems. It is found that the coefficients of enhancing feedback control and dislocated feedback control may be smaller than those of ordinary feedback control, so, the complexity and cost of the system control are reduced. Numerical simulations are given to illustrate the effectiveness of the proposed controllers.     

Control and adaptive modified function projective synchronization of Liu chaotic dynamical system

In this work, the feedback control method is proposed to control the behaviour of Liu chaotic dynamical system. The controlled system is stable under some conditions on the parameters of the system determined by Routh-Hurwitz criterion. This paper also presents the adaptive modified function projective synchronization (AMFPS) between two identical Liu chaotic dynamical systems. Based on the Lyapunov stability theorem, adaptive control laws are designed to achieving the AMFPS. Finally, some numerical simulations are obtained to validate the proposed methods.     

Feedback control and hybrid projective synchronization of a fractional-order Newton-Leipnik system

In this work, stability analysis of the fractional-order Newton-Leipnik system is studied by using the fractional Routh-Hurwitz criteria. The fractional Routh-Hurwitz conditions are used to control chaos in the proposed fractional-order system to its equilibria. Based on the fractional Routh-Hurwitz conditions and using specific choice of linear feedback controllers, it is shown that the Newton-Leipnik system is controlled to its equilibrium points. Moreover, the theoretical basis of hybird projective synchronization of commensurate and incommensurate fractional-order Newton-Leipnik systems is investigated. Based on the stability theorems of fractional-order systems, the controllers for hybrid projective synchroniztion are derived. Numerical results show the effectiveness of the theoretical analysis.     

Circuit realization,chaos synchronization and estimation of parameters of a hyperchaotic system with unknown parameters

In this article, the adaptive chaos synchronization technique is implemented by an electronic circuit and applied to the hyperchaotic system proposed by Chen et al. We consider the more realistic and practical case where all the parameters of the master system are unknowns. We propose and implement an electronic circuit that performs the estimation of the unknown parameters and the updating of the parameters of the slave system automatically, and hence it achieves the synchronization. To the best of our knowledge, this is the first attempt to implement a circuit that estimates the values of the unknown parameters of chaotic system and achieves synchronization. The proposed circuit has a variety of suitable real applications related to chaos encryption and cryptography. The outputs of the implemented circuits and numerical simulation results are shown to view the performance of the synchronized system and the proposed circuit.     

Tracking control and generalized projective synchronization of a class of hyperchaotic system with unknown parameter and disturbance

In this paper, the tracking control and generalized projective synchronization of a class of hyperchaotic system with unknown parameter and disturbance are investigated. Based on the LaSalle’s invariant set theorem, a robust adaptive controller is contrived to acquire tracking control and generalized projective synchronization and parameter identification simultaneously. It is proved theoretically that the proposed scheme can allow us to drive the hyperchaotic system to any desired reference signals, including hyperchaotic signals, chaotic signals, periodic orbits or fixed value by the given scaling factor. The presented simulation results further demonstrate that the proposed method is effective and robust.     

Modified generalized projective synchronization of a new fractional-order hyperchaotic system and its application to secure communication

This paper presents a new fractional-order hyperchaotic system. The chaotic behaviors of this system in phase portraits are analyzed by the fractional calculus theory and computer simulations. Numerical results have revealed that hyperchaos does exist in the new fractional-order four-dimensional system with order less than 4 and the lowest order to have hyperchaos in this system is 3.664. The existence of two positive Lyapunov exponents further verifies our results. Furthermore, a novel modified generalized projective synchronization (MGPS) for the fractional-order chaotic systems is proposed based on the stability theory of the fractional-order system, where the states of the drive and response systems are asymptotically synchronized up to a desired scaling matrix. The unpredictability of the scaling factors in projective synchronization can additionally enhance the security of communication. Thus MGPS of the new fractional-order hyperchaotic system is applied to secure communication. Computer simulations are done to verify the proposed methods and the numerical results show that the obtained theoretic results are feasible and efficient.     

Stability and Hopf bifurcation analysis of novel hyperchaotic system with delayed feedback control

In this article, a novel four dimensional autonomous nonlinear systezm called hyperchaotic Rikitake system is proposed. Basic properties of the new system are investigated and the complex dynamical behaviors, such as time series, bifurcation diagram, and Lyapunov exponents are analyzed by dynamic analysis approaches. To control the new hyperchaotic system, the delayed feedback control is introduced. Regarding the time delay as a bifurcation parameter, stability and bifurcations with respect to time delay are investigated. Conditions assuring the existence of Hopf bifurcation and the distribution of roots to the associated characteristic equation are investigated by utilizing the polynomial theorem. Besides, the Hopf bifurcation is proved to occur when the bifurcation parameter (time delay) crosses through derived critical value. Finally, numerical simulations are provided to prove the consistence with the derived theoretical results. © 2015 Wiley Periodicals, Inc. Complexity 21: 180–193, 2016     

Control of a novel fractional hyperchaotic system using a located control method

Fractional order dynamics and chaotics systems have been recently combined, yielding interesting behaviours. In this paper, a novel integer order hyperchaotic system is considered. Then, a fractional order hyperchaotic representation of said system is proposed using a natural fractionalization. Two different linear control methodologies to deal with the complexity which introduce such systems are proposed. Those methods are able to modify the hyperchaotic behaviour of the system and force it to move towards a fixed point; i.e. steady state. These approaches give a general framework for taming such complex systems using simple linear controllers. The main tools for analysing the controlled system are Matignon stability criterion and RouthHurwitz test. Using a reliable numerical simulation, the designed system is simulated to verify the theoretical analysis.     

Ultimate bound sets of a hyperchaotic system and its application in chaos synchronization

In this article, we investigate globally exponentially attractive sets and chaos synchronization for a hyperchaotic system, namely, Lorenz–Stenflo system. For this system, two ellipsoidal globally exponentially attractive sets are derived based on generalized Lyapunov function theory and the extremum principle of function. Furthermore, we propose linear feedback control with a one, two, three, and four inputs to realize globally exponential synchronization of two four‐dimesional hyperchaotic systems using inequality techniques. Numerical simulations are presented to show the effectiveness of the proposed synchronization scheme. © 2014 Wiley Periodicals, Inc. Complexity 20: 30–44, 2015     

Generalized function projective (lag, anticipated and complete) synchronization between two different complex networks with nonidentical nodes

Generalized function projective (lag, anticipated and complete) synchronization between two different complex networks with nonidentical nodes is investigated in this paper. Based on Barbalat’s lemma, some sufficient synchronization criteria are derived by applying the nonlinear feedback control. Although previous work studied function projective synchronization on complex dynamical networks, the dynamics of the nodes are coupled partially linear chaotic systems. In our work, the dynamics of the nodes of the complex networks are any chaotic systems without the limitation of the partial linearity. In addition, each network can be undirected or directed, connected or disconnected, and nodes in either network may have identical or different dynamics. The proposed strategy is applicable to almost all kinds of complex networks. Numerical simulations further verify the effectiveness and feasibility of the proposed synchronization method. Numeric evidence shows that the synchronization rate is sensitively influenced by the feedback strength, the time delay, the network size and the network topological structure.     

Existence and Exponential Stability for a Euler-Bernoulli Beam Equation with Memory and Boundary Output Feedback Control Term

In this paper, we consider the Euler-Bernoulli beam equation with memory and boundary output feedback control term. We prove the existence of solutions using the Galerkin method and then investigate the exponential stability of solutions by using multiplier technique. This work was supported by grant number KRF-2005-202-C00030 from the Korea Research Foundation. This work was supported by National Institute for Mathematical Sciences.     

Control of AMIRA’s ball and beam system via improved fuzzy feedback linearization approach

This paper first studies the tracking and almost disturbance decoupling problem of nonlinear AMIRA’s ball and beam system based on the feedback linearization approach and fuzzy logic control. The main contribution of this study is to construct a controller, under appropriate conditions, such that the resulting closed-loop system is valid for any initial condition and bounded tracking signal with the following characteristics: input-to-state stability with respect to disturbance inputs and almost disturbance decoupling, i.e., the influence of disturbances on the L₂ norm of the output tracking error can be arbitrarily attenuated by changing some adjustable parameters. One example, which cannot be solved by the first paper on the almost disturbance decoupling problem, is proposed in this paper to exploit the fact that the tracking and the almost disturbance decoupling performances are easily achieved by our proposed approach. The simulation results show that our proposed approach has achieved the almost disturbance decoupling performance perfectly.     

Painlevé property, soliton-like solutions and complexitons for a coupled variable-coefficient modified Korteweg-de Vries system in a two-layer fluid model

As a model derived from a two-layer fluid system which describes the atmospheric and oceanic phenomena, a coupled variable-coefficient modified Korteweg-de Vries system is concerned in this paper. With the help of symbolic computation, its integrability in the Painlevé sense is investigated. Furthermore, Hirota’s bilinear method is employed to construct the bilinear forms through the dependent variable transformations, and soliton-like solutions and complexitons are derived. Finally, effects of variable coefficients are discussed graphically, and it is concluded that the variable coefficients control the propagation trajectories of solitons and complexitons.