Lorenz-Stenflo Chaotic System Research Articles

Combination Control of Chaotic Systems

In this paper, the tracking control of a combination of two, three, four and five chaotic systems is achieved for the 2-D Duffing oscillator, 3-D Lorenz chaotic system and 4-D Lorenz–Stenflo chaotic system via recursive backstepping technique. The combination tracking control could find applications in situations where the total output of several chaotic systems is required in a desired form. Numerical simulations are carried out to validate the schemes.

Qualitative behaviors of the high-order Lorenz-Stenflo chaotic system arising in mathematical physics describing the atmospheric acoustic-gravity waves

The boundedness of chaotic systems plays an important role in investigating the stability of the equilibrium, estimating the Lyapunov dimension of attractors, the Hausdorff dimension of attractors, the existence of periodic solutions, chaos control, and chaos synchronization. However, as far as the authors know, there are only a few papers dealing with bounds of high-order chaotic systems due to their complex algebraic structure. To sort this out, in this paper, we study the bounds of a high-order Lorenz-Stenflo system arising in mathematical physics. Based on Lyapunov stability theory, we show that there exists a globally exponential attractive set for this system. The innovation of the paper is that we not only prove that this system is globally bounded for all the parameters, but also give a family of mathematical expressions of global exponential attractive sets of this system with respect to its parameters. We also study some other dynamical characteristics of this chaotic system such as invariant sets and chaotic behaviors. To justify the theoretical analysis, we carry out detailed numerical simulations.

Control of Lorenz-StenfloChaotic System via Takagi-SugenoFuzzy Model Based on Linear Matrix Inequality

This paper investigates the chaos control problem for Lorenz-Stenflo chaotic system with uncertain parameters. Based on the interval matrix theory, the Takagi-Sugeno(T-S)fuzzy model is applied to the Lorenz-Stenflo chaotic system with uncertain parameters.Based on theLyapunov stability theorem, the sufficient stabilitycondition for the Lorenz-Stenflo chaotic system is presented as a set of linear matrix inequality (LMI) forthe first time and the strict mathematical norms of LMI are given. Then the controller feedback gain can be obtained by solving a set of linear matrix inequality. Finally, simulation results for the Lorenz-Stenflo chaotic system are provided to illustrate the effectiveness of the proposed scheme.

Passivity-based chaos control and synchronization of the four dimensional Lorenz-Stenflo system via one input

This paper discusses chaos control and synchronization of the four dimensional Lorenz-Stenflo system based on passive control technique. Using passive theory, chaos control and synchronization of a four dimensional Lorenz-Stenflo chaotic system are realized with one input. The designed controllers ensure the stability of the controlled system and error dynamical system between two identical Lorenz-Stenflo systems. Also, the controllers ensure that the controlled system and error dynamical system converge to zero equilibrium. Numerical simulations show that the proposed method is effective for the Lorenz-Stenflo system.

One input control of exponential synchronization for a four-dimensional chaotic system

The purpose of this paper is to concern the exponential synchronization of the four-dimensional (4D) Lorenz–Stenflo chaotic system by utilizing one input linear feedback control. There are two main results provided. Firstly, for two identical chaotic systems, two types of one input controllers with single state linear feedback applied to the slave system are introduced. Then, these one input linear feedback controllers associated with the corresponding system parameter updated rules are also developed for two non-identical chaotic systems. Sufficient conditions are proven based on the exponential stability and numerical simulations are proposed to demonstrate the effectiveness of the present schemes.

Two fuzzy control schemes for Lorenz-Stenflo chaotic system

In this paper, two effective fuzzy control schemes, i.e., fuzzy feedback control method and adaptive fuzzy control method are introduced to suppress the state variables of the Lorenz-Stenflo chaotic system (LSCS) to its equilibrium point. For the first fuzzy control scheme, with the abundant modeling capability of the T-S fuzzy model, the LSCS can be decomposed into some local linear systems, which makes it very convenient to use the fuzzy feedback control method to analyze it. Based on the Lyapunov stability theorem, a criterion is also derived to guarantee the controlled LSCS is robust stable at the equilibrium point, even if there exist external perturbations. For the second fuzzy control scheme, fuzzy logic systems areexploited to approximate the nonlinear functions. Moreover, an adaptive technique is employed to construct an effective fuzzy controller, which can drive all state variables into a rather small neighborhood of its equilibrium point. By choosing a group of suitable parameters, the controlled system can approach the equilibrium point with high precision. Two numerical simulations are presented to demonstrate the effectiveness and feasibility of the two fuzzy control schemes.

Backstepping Control and Synchronization for 4-D Lorenz-Stenflo Chaotic System with Single Input

In this paper, a backstepping design is proposed to achieve stabilization and synchronization for the Lorenz-Stenflo (LS) chaotic system. The proposed method is a recursive Lyapunov-based scheme and provides a systematic procedure to design stabilizing controllers. The proposed controller enables stabilization of the chaotic motion and synchronization of two identical LS chaotic systems using only a single control input. Numerical simulations are presented to validate the proposed method.

Time-delayed generalized projective synchronization of piecewise chaotic system with unknown parameters

The time-delayed generalized projective synchronization problem for a class of piecewise modified Lorenz-Stenflo chaotic system with unknown parameters is discussed. The adaptive nonlinear feedback controller and the parameter update rules are designed based on the Lyapunov stability theory, and the feedback gain can be adjusted adaptively according to the error values and the states. The method can make all full states asymptotically generalized projected synchronization and identify the real-time drive system and the time-delayed response system’s unknown parameters. Results of simulation verified the realistic feasibility and effectiveness of the proposed method to the time-delayed generalized projective synchronization of the piecewise chaotic system.