A New Chaotic System with Line of Equilibria: Dynamics, Passive Control and Circuit Design

A new chaotic system with line equilibrium is introduced in this paper. This system consists of five terms with two transcendental nonlinearities and two quadratic nonlinearities. Various tools of dynamical system such as phase portraits, Lyapunov exponents, Kaplan-Yorke dimension, bifurcation diagram and Poincarè map are used. It is interesting that this system has a line of fixed points and can display chaotic attractors. Next, this paper discusses control using passive control method. One example is given to insure the theoretical analysis. Finally, for the  new chaotic system, An electronic circuit for realizing the chaotic system has been implemented. The numerical simulation by using MATLAB 2010 and implementation of circuit simulations by using MultiSIM 10.0 have been performed in this study

Hybrid Stability Checking Method for Synchronization of Chaotic Fractional-Order Systems

A hybrid stability checking method is proposed to verify the establishment of synchronization between two hyperchaotic systems. During the design stage of a synchronization scheme for chaotic fractional-order systems, a problem is sometimes encountered. In order to ensure the stability of the error signal between two fractional-order systems, the arguments of all eigenvalues of the Jacobian matrix of the erroneous system should be within a region defined in Matignon’s theorem. Sometimes, the arguments depend on the state variables of the driving system, which makes it difficult to prove the stability. We propose a new and efficient hybrid method to verify the stability in this situation. The passivity-based control scheme for synchronization of two hyperchaotic fractional-order Chen-Lee systems is provided as an example. Theoretical analysis of the proposed method is validated by numerical simulation in time domain and examined in frequency domain via electronic circuits

Controlling Hyperchaotic Finance System with Combining Passive and Feedback Controllers

In this paper, a novel control method that combines passive, linear feedback, and dislocated feedback control methods is proposed and applied to the control of the four-dimensional hyperchaotic finance system which has been introduced and controlled with the linear feedback and speed feedback control methods by Yu, Cai, and Li (2012). The stability of the hyperchaotic finance system at its equilibrium points is ensured on the basis of a Lyapunov function. Computer simulations are used for verifying all the theoretical analyses visually. In the simulations, the proposed control method is also compared with the speed feedback and linear feedback control methods to observe its effectiveness. Finally, the comparative findings are discussed

Model Reference Control of Hyperchaotic Systems

We have applied a famous engineering method, called model reference control, to control hyperchaos. We have proposed a general description of the hyperchaotic system and its reference system. By using the Lyapunov stability theorem, we have obtained the expression of the controller. Four examples for the both certain case and the uncertain case show that our method is very effective for controlling hyperchaotic systems with both certain parameters and uncertain parameters

Adaptive Second-Order Sliding Mode Algorithm-Based Modified Function Projective Synchronization of Uncertain Hyperchaotic Systems

This article proposes a synchronization technique for uncertain hyperchaotic systems in the modified function projective manner using integral fast terminal sliding mode (I-FTSM) and adaptive second-order sliding mode algorithm. The new I-FTSM manifolds are introduced with the aim of having the fast convergence speed. The proposed continuous controller not only results in the robustness and high-accuracy synchronization in the presence of unknown external disturbances and/or model uncertainties but also helps alleviating the chattering effect significantly. Numerical simulation results are provided to illustrate the effectiveness of the proposed control design technique and verify the theoretical analysis

Synchronization of chaos in nonlinear finance system by means of sliding mode and passive control methods: A comparative study

In this paper, two different control methods, namely sliding mode control and passive control, are investigated for the synchronization of two identical chaotic finance systems with different initial conditions. Based on the sliding mode control theory, a sliding surface is determined. A Lyapunov function is used to prove that the passive controller provides global asymptotic stability of the system. Numerical simulations validate the synchronization of chaotic finance systems with the proposed sliding mode and passive control methods. The synchronization performance of these two methods is compared and discussed

Robust synchronization of fractional-order unified chaotic systems via linear control

AbstractA new scheme for accomplishing synchronization between two fractional-order unified chaotic systems is proposed in this paper. The scheme does not require that the nonlinear dynamics of the synchronization error system must be eliminated. Moreover, the parameter of the systems does not have to be known. A controller is a linear feedback controller, which is simple in implementation. It is designed based on an LMI condition. The LMI condition guarantees that the synchronization between the slave system and the master system is achieved. Numerical simulations are performed to demonstrate the effectiveness of the proposed scheme