Synchronization of a four-dimensional energy resource system via linear control

Synchronization of a four-dimensional energy resource system is investigated. Four linear control schemes are proposed to synchronize energy resource chaotic system via the back-stepping method. We use simpler controllers to realize a global asymptotical synchronization. In the first three 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 fourth scheme, the synchronization condition is obtained by numerical method, in which only one state variable controller is contained. Finally, four numerical simulation examples are performed to verify these results.     

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.     

Synchronization of chaotic systems using feedback controller: An application to Diffie–Hellman key exchange protocol and ElGamal public key cryptosystem

In this paper, designing an appropriate linear and nonlinear feedback control, the two identical integer order chaotic systems are synchronized by analytically and numerically. It has been realizing that, synchronization using linear feedback control method is efficient than nonlinear feedback control method due to the less computational complexity and the synchronization error. ElGamal public key cryptosystem is described through the proposed Diffie–Hellman key exchange protocol based on the synchronized chaotic systems using linear feedback control and their security are analyzed. The numerical simulations are given to validate the correctness of the proposed synchronization of chaotic systems and the ElGamal cryptosystem.     

Global asymptotical synchronization of chaotic neural networks by output feedback impulsive control: An LMI approach

In this paper, impulsive control for master–slave synchronization schemes consisting of identical chaotic neural networks is studied. Impulsive control laws are derived based on linear static output feedback. A sufficient condition for global asymptotic synchronization of master–slave chaotic neural networks via output feedback impulsive control is established, in which synchronization is proven in terms of the synchronization errors between the full state vectors. An LMI-based approach for designing linear static output feedback impulsive control laws to globally asymptotically synchronize chaotic neural networks is discussed. With the help of LMI solvers, linear output feedback impulsive controllers can be easily obtained along with the bounds of the impulsive intervals for global asymptotic synchronization. The method is finally illustrated by numerical simulations.     

Projective lag synchronization in chaotic systems

In this paper, a new projective lag synchronization is proposed, where a driven chaotic system synchronizes the past state of the driver up to a scaling factor α. An active control method is employed to design a controller to achieve the global synchronization of two identical chaotic systems. Based on Lyapunov stability theorem, a sufficient condition is then given for the asymptotical stability of the null solution of an error dynamics. The effectiveness of the proposed schemes is verified via numerical simulations.     

Dynamics and adaptive synchronization of the energy resource system

This paper discusses a new energy resource chaotic system. It investigates basically dynamical behaviors of this new system. It also addresses the synchronization problem of two energy resource systems in the presence of different unknown system parameters. Based on Lyapunov stability theory, an adaptive control law is derived to make the states of two energy resource systems with different unknown system parameters asymptotically synchronized. Numerical simulations are given to validate the synchronization approach.     

Lag synchronizing chaotic system based on a single controller

Lag synchronization of chaotic system is investigated. Three kinds of schemes are proposed to lag synchronize Chen chaotic system. All the three schemes need only a single controller to realize lag synchronization. Especially in the last two schemes, only one state variable is contained in controller, which is of important significance on using chaos lag synchronization for applications. Finally numerical simulations are provided to show the effectiveness of the developed methods.     

Linear generalized synchronization between two complex networks

In this paper, the linear generalized synchronization between two complex networks is investigated. Based on the Lyapunov stability theory, a simple criterion for linear generalized synchronization between two networks with the same connection topologies is attained by using the nonlinear control method, which can widen the application range of the generalized synchronization methods. The feasibility of the proposed scheme is proved in theory and numerical simulations further demonstrate the effectiveness of it.     

Adaptive feedback controller for projective synchronization

Due to the unpredictability of the scaling factor of projective synchronization in coupled partially linear systems, it is hard to know for sure the terminal state of the synchronized dynamics. In this paper, a simple adaptive linear feedback control method is proposed for controlling the scaling factor onto a desired value, based on the invariance principle of differential equations. Firstly, we prove the synchronizability of the proposed simple adaptive projective synchronization control method from the viewpoint of mathematics. Then, two numerical examples are presented to illustrate the applications of the derived results. Finally, we propose a communication scheme based on the adaptive projective synchronization of the Lorenz chaotic system. Numerical simulation shows its feasibility.     

Conservative finite difference methods for fractional Schrödinger–Boussinesq equations and convergence analysis

In this paper, two conservative finite difference schemes for fractional Schrödinger–Boussinesq equations are formulated and investigated. The convergence of the nonlinear fully implicit scheme is established via discrete energy method, while the linear semi‐implicit scheme is analyzed by means of mathematical induction method. Our schemes are proved to preserve the total mass and energy in discrete level. The numerical results are given to confirm the theoretical analysis.     

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     

Exponential generalized synchronization of uncertain coupled chaotic systems by adaptive control

In this paper, the exponential generalized synchronization for a class of coupled systems with uncertainties is defined. A novel and powerful method is proposed to investigate the generalized synchronization based on the adaptive control technique. According to the Lyapunov stability theory, rigorous proof is given for the exponential stability of error system. In comparison with previous schemes, the presented method shortens the synchronization time and is more applicable in practice. Besides, it is shown that the synchronization effect is robust against the uncertain factors. Some typical chaotic and hyper-chaotic systems are taken as examples to illustrate above approach. The corresponding numerical simulations are demonstrated to verify the effectiveness of proposed method.     

Cascades-based synchronization of hyperchaotic systems: Application to Chen systems

We discuss the cascaded-based controlled synchronization method for hyperchaotic systems. The control approach is based on analysis tools for cascaded time-varying systems. That is, the closed-loop system takes the form of two subsystems which are interconnected in a manner that the state of one system enters into another but without feedback loop. The advantage of such construction is that the controller is largely simplified relative to other design methods such as backstepping. We apply the method to Chen’s hyperchaotic system and show that global synchronization is achieved via linear control. Also, we assume that only three instead of four control inputs are available. The method is tested in numerical simulations.     

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.     

Global synchronization criterion and adaptive synchronization for new chaotic system

This paper proposes two schemes of synchronization of two four-scorll chaotic attractor, a simple global synchronization and adaptive synchronization in the presence of unknown system parameters. Based on Lyapunov stability theory and matrix measure, a simple generic criterion is derived for global synchronization of four-scorll chaotic attractor system with a unidirectional linear error feedback coupling. This methods are applicable to a large class of chaotic systems where only a few algebraic inequalities are involved. Numerical simulations are presented to show the effectiveness of the proposed chaos synchronization method.     

A new secured transmission scheme based on chaotic synchronization via smooth adaptive unknown-input observers

We present a new scheme for the secured transmission of information based on master–slave synchronization of chaotic systems, using unknown-input observers. Our approach improves upon state-of-the-art schemes by being compatible with information of relatively large amplitude while improving security against intruders through an intricate encryption system. In addition, our approach is robust to channel noise. The main idea is to separate the encryption and synchronization operations by using two cascaded chaotic systems in the transmitter. Technically, the scheme is based on smooth adaptive unknown-input observers; these have the advantage to estimate the (master) states and to reconstruct the unknown inputs simultaneously. The performance of the communication system is illustrated in numerical simulation.     

Synchronization and anti-synchronization coexist in Chen–Lee chaotic systems

This study demonstrates that synchronization and anti-synchronization can coexist in Chen–Lee chaotic systems by direct linear coupling. Based on Lyapunov’s direct method, a linear controller was designed to assure that two different types of synchronization can simultaneously be achieved. Further, the hybrid projective synchronization of Chen–Lee chaotic systems was studied using a nonlinear control scheme. The nonlinear controller was designed according to the Lyapunov stability theory to guarantee the hybrid projective synchronization, including synchronization, anti-synchronization, and projective synchronization. Finally, numerical examples are presented in order to illustrate the proposed synchronization approach.     

Novel energy stable schemes for Swift-Hohenberg model with quadratic-cubic nonlinearity based on the H−1-gradient flow approach

The Swift-Hohenberg model is a very important phase field crystal model which can be described many crystal phenomena. This model with quadratic-cubic nonlinearity based on the H^??1-gradient flow approach is a sixth-order system which satisfies mass conservation and energy dissipation law. The negative energy of this model will bring huge difficulties to energy stability for many existing approaches. In this paper, we consider two linear, second-order and unconditionally energy stable schemes by linear invariant energy quadratization (LIEQ) and modified scalar auxiliary variable (MSAV) approaches. These two schemes will be effective for all negative E₁. Furthermore, we proved that all the semi-discrete schemes are unconditionally energy stable with respect to a modified energy. Finally, we present various 2D numerical simulations to demonstrate the stability and accuracy.

     

Numerical simulation of the boundary exact control for the system of linear elasticity

The problem of computing numerically the boundary exact control for the system of linear elasticity in two dimensions is addressed. A numerical method which has been recently proposed in [P. Pedregal, F. Periago, J. Villena, A numerical method of local energy decay for the boundary controllability of time-reversible distributed parameter systems. Stud. Appl. Math. 121 (1) (2008) 27–47] is implemented. Two cases are considered: first, a rectangular domain with Dirichlet controls acting on two adjacent edges, and secondly, a circular domain with Neumann controls distributed along the whole boundary.