Bifurcations, chaos and synchronization in ADVP circuit with parallel resistor

In this work we investigate the dynamical behaviors of Van der Pol-Duffing circuit (ADVP) with parallel resistor. The model is described by a continuous-time three dimensional autonomous system. The stability conditions of the equilibria are analyzed. The existence of periodic solutions and their stabilities about the node equilibrium point of the system are studied by using Hopf's theorem and Hsü and Kazarinoff theorem. Lyapunov spectrum is calculated for the proposed system. Adaptive synchronization using backstepping design is applied successfully to the system. Chaotic behaviors and the efficiency of the synchronization method are verified by numerical simulations.     

Dynamical properties and chaos synchronization of improved Colpitts oscillators

In this paper, the dynamics and synchronization of improved Colpitts oscillators designed to operate in ultrahigh frequency range are considered. The model is described by a continuous time four-dimensional autonomous system with an exponential nonlinearity. The system is integrated numerically and various bifurcation diagrams and corresponding graphs of largest 1D Lyapunov exponent are plotted to summarize different scenarios leading to chaos. It is found that the oscillator moves from the state of fixed point motion to chaos via the usual paths of period-doubling, intermittency and interior crisis routes when monitoring the bias (i.e. power supply) in tiny ranges. In order to promote chaos-based synchronization designs of this type of oscillators, a synchronization strategy based upon the design of a nonlinear state observer is successfully adapted. The suggested approach enables synchronization to be achieved via a scalar transmitted signal which represents a suitable feature for communication applications. Numerical simulations are performed to demonstrate the effectiveness and feasibility of the proposed technique.     

Suppressing chaos in a simplest autonomous memristor-based circuit of fractional order by periodic impulses

In this paper, the chaotic behavior of a simplest autonomous memristor-based circuit of fractional order is suppressed by periodic impulses applied to one or several state variables. The circuit consists of two passive linear elements, a capacitor and an inductor, as well as a nonlinear memristive element. It is shown that by applying a sequence of adequate (identical or different) periodic impulses to one or several variables, the chaotic behavior can be suppressed. Impulse values and control timing are determined numerically, based on the bifurcation diagram with impulses as bifurcation parameters. Empirically, the probability to have a reasonably wide range of impulses to suppress chaos is quite large, ensuring that chaos suppression can be implemented, as demonstrated by several examples presented.     

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.     

Dynamical analysis,feedback control and synchronization of Liu dynamical system

Dynamical behaviors of Liu system is studied using Routh–Hurwitz criteria, Center manifold theorem and Hopf bifurcation theorem. Periodic solutions and their stabilities about the equilibrium points are studied by using Hsü & Kazarinoff theorem. Linear feedback control techniques are used to stabilize and synchronize the chaotic Liu system.     

Dynamical analysis and chaos control in a heterogeneous Kopel duopoly game

A dynamic of a nonlinear Kopel duopoly game with heterogeneous players is presented. By assuming two heterogeneous players where one player use naive expectation whereas the other employs a technique of adaptive. The stability conditions of equilibrium points are analyzed. Numerical simulations are used to show bifurcation diagrams, phase portraits and sensitive dependence on initial conditions. The chaotic behavior of the game has been controlled by using feedback control method.     

Dynamic behavior and chaos control in a complex Riccati-type map

This paper is devoted to analyze the dynamic behavior of a Riccatitype map with complex variables and complex parameters. Fixed points and their asymptotic stability are studied. Lyapunov exponent is computed to indicate chaos. Bifurcation and chaos are discussed. Chaotic behavior of the map has been controlled by OGY feedback control method.     

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.     

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.     

Universal chaos synchronization control laws for general quadratic discrete systems

This paper addresses the reliable universal synchronization problem between two coupled chaotic quadratic discrete systems. A general nonlinear control method of synchronization for coupled 2D and 3D quadratic dynamical systems in discrete-time is proposed. The proposed synchronization method is based on universal controllers. The synchronization results are derived theoretically using active control method and Lyapunov stability theory. Numerical simulations are performed to assess the performance of the presented analysis.     

On chaos control and synchronization of the commensurate fractional order Liu system

In this work, we study chaos control and synchronization of the commensurate fractional order Liu system. Based on the stability theory of fractional order systems, the conditions of local stability of nonlinear three-dimensional commensurate fractional order systems are discussed. The existence and uniqueness of solutions for a class of commensurate fractional order Liu systems are investigated. We also obtain the necessary condition for the existence of chaotic attractors in the commensurate fractional order Liu system. The effect of fractional order on chaos control of this system is revealed by showing that the commensurate fractional order Liu system is controllable just in the fractional order case when using a specific choice of controllers. Moreover, we achieve chaos synchronization between the commensurate fractional order Liu system and its integer order counterpart via function projective synchronization. Numerical simulations are used to verify the analytical results.     

Projective lag synchronization of spatiotemporal chaos via active sliding mode control

This paper investigates projective lag synchronization of spatiotemporal chaos with disturbances. A control scheme is designed via active sliding mode control. The synchronization of spatiotemporal chaos between a drive system and a response system with disturbances and time-delay is implemented by adding the active sliding mode controllers. The control law is applied to two identical spatiotemporal Gray-Scott systems. Numerical results demonstrate the feasibility and the effectiveness of the proposed approach.     

Global chaos synchronization of new chaotic system using linear active control

Chaos synchronization is a procedure where one chaotic oscillator is forced to adjust the properties of another chaotic oscillator for all future states. This research paper studies and investigates the global chaos synchronization problem of two identical chaotic systems and two non‐identical chaotic systems using the linear active control technique. Based on the Lyapunov stability theory and using the linear active control technique, the stabilizing controllers are designed for asymptotically global stability of the closed‐loop system for both identical and non‐identical synchronization. Numerical simulations and graphs are imparted to justify the efficiency and effectiveness of the proposed scheme. All simulations have been done by using mathematica 9. © 2014 Wiley Periodicals, Inc. Complexity 21: 379–386, 2015     

Chaos control and chaos synchronization for multi-scroll chaotic attractors generated using hyperbolic functions

In this paper, we design a series of chaotic systems that can generate one-directional, two-directional and three-directional multi-scroll chaotic attractors. Then, based upon the properties of these chaotic systems, we construct appropriate Lyapunov functions and design simple linear feedback controls to globally exponentially stabilize and synchronize these chaotic systems. Numerical simulation results are also presented to show the applicability of the proposed control laws.     

Chaos and chaos synchronization for a non-autonomous rotational machine systems

Chaos and chaos synchronization of the centrifugal flywheel governor system are studied in this paper. By mechanics analyzing, the dynamical equation of the centrifugal flywheel governor system is established. Because of the non-linear terms of the system, the system exhibits both regular and chaotic motions. The characteristic of chaotic attractors of the system is presented by the phase portraits and power spectra. The evolution from Hopf bifurcation to chaos is shown by the bifurcation diagrams and a series of Poincaré sections under different sets of system parameters, and the bifurcation diagrams are verified by the related Lyapunov exponent spectra. This letter addresses control for the chaos synchronization of feedback control laws in two coupled non-autonomous chaotic systems with three different coupling terms, which is demonstrated and verified by Lyapunov exponent spectra and phase portraits. Finally, numerical simulations are presented to show the effectiveness of the proposed chaos synchronization scheme.     

An adaptive feedback control for chaos synchronization of nonlinear systems with different order

This paper deals with the chaotic synchronization of fourth-order system and second driven oscillators. Such a problem is related to the synchronization of strictly different chaotic systems. We show that the dynamical evolution of second-order driven oscillators can be synchronized with the projection on canonical planes of a fourth-order chaotic system. In this sense, it is said that the synchronization is achieved in reduced-order. Based on the Lyapunov approach, the adaptation law is determined to tune the controller gain vector in order to track a predetermined linearizing feedback control. An application to secure chaotic communication is also discussed. Numerical simulations are presented to demonstrate the efficiency of the proposed synchronization and secure communication schemes.     

Bifurcation,chaos, and their control in a wheelset model

In this paper, we present an improved wheelset motion model with two degrees of freedom and study the dynamic behaviors of the system including the symmetry, the existence and uniqueness of the solution, continuous dependence on initial conditions, and Hopf bifurcation. The dynamic characteristics of the wheelset motion system under a nonholonomic constraint are investigated. These results generalize and improve some known results about the wheelset motion system. Meanwhile, based on multiple equilibrium analysis, calculation of Lyapunov exponents and Poincaré section, the chaotic behaviors of the wheelset system are discussed, which indicates that there are more complex dynamic behaviors in the railway wheelset system with higher order terms of Taylor series of trigonometric functions. This paper has also realized the chaos control and bifurcation control for the wheelset motion system by adaptive feedback control method and linear feedback control. The results show that the chaotic wheelset system and bifurcation wheelset system are all well controlled, whether by controlling the yaw angle and the lateral displacement or only by controlling the yaw angle. Numerical simulations are carried out to further verify theoretical analyses.     

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.