Partial Region Stability Theory Research Articles

Generalized synchronization of regulate seizures dynamics in partial epilepsy with fractional-order derivatives

Abstract The dynamical behavior and the synchronization of epileptic seizure dynamics, with fractional-order derivatives, is studied in this paper. Knowing that the dynamical properties of ictal electroencephalogram signal recordings during experiments displays complex nonlinear behaviors, we analyze the system from chaos theory point of view. Based on stability analysis, the system presents three equilibrium points with two of them unstable. Moreover, the system reveals attractor points from the phase portrait analysis. In addition, the largest Lyapunov exponent displays positive values after a given period of time. These observations characterize a chaotic behavior of epileptic seizure dynamics. Therefore, based on the Ge-Yao-Chen partial region stability theory, the synchronization of the system is achieved and simulations prove that the control technique is very efficient. Further studies based on phase synchronization show that we are able to force infected population of neurons by epilepsy into synchronization with uninfected one through a coupling constant. In addition, based on the phase locking value time evolution (phase synchronization) of the system, we realize that fractional-order derivative induces quick synchronization compared to integer order derivative. These results might be very interesting from the medical point of view, because by applying the proposed control method, one may be able to regulate (or reduce) seizure amplitude which, if kindly implemented in practice, will provide excellent therapeutic solution to drug resistant patients with epilepsy.

Generalized synchronization of the extended Hindmarsh–Rose neuronal model with fractional order derivative

In the present paper, the synchronization of the extended Hindmarsh–Rose neuronal model with fractional order derivative is presented in detail taking into consideration the effects of the very slow intracellular exchange of calcium ion (Ca2+) occurring between cytoplasm and its store. That is, we approach the synchronization by the Ge–Yao–Chen partial region stability theory. The method of resolution used is the powerful Adam–Bashforth–Moulton method which converges quickly to the exact solution. Numerical simulations of the error dynamics proved the effectiveness of the control method. The synchronization time is evaluated for different orders of the derivative and its analysis shows that, it is closely related to the order of the derivative and that fractional order induces quick synchronization compared to integer order.

Chaos generalized synchronization of coupled Mathieu-Van der Pol and coupled Duffing-Van der Pol systems using fractional order-derivative

In the present paper, the synchronization by the Ge-Yao-Chen (GYC) partial region stability theory of chaotic Mathieu-Van der Pol and chaotic Duffing-Van der Pol systems with fractional order-derivative is proposed. Numerical simulations show that this synchronization technique is very effective and it turns out that the fractional order-derivative induces quick synchronization compared to integer order-derivative of these systems. In order to bring out the chaotic behavior of these systems either with fractional or with integer order-derivative, we simulate their phase portraits and the Lyapunov exponent. Moreover, we provide in this work an approximated solution to both systems to show that the solution of such a system can be represented as a simple power-series function. Furthermore, the representation of the error dynamics with respect to the time before and after the control action approves the effectiveness of the control method and proves the possibility of stabilization and controllability of chaotic systems with an appropriate. Furthermore, the synchronization of the fractional Mathieu-Van der Pol system using the fractional Duffing-Van der Pol system is simulated.

Nonlinear Dynamic Analysis and Synchronization of Four-Dimensional Lorenz-Stenflo System and Its Circuit Experimental Implementation

Recently many chaotic systems’ circuits are designed to generate phenomenon of chaos signals. The ability to synchronize chaotic circuits opens a great number of ways to use them in application signals masking. In this paper, first a new nonlinear chaotic dynamical system had be design, analyze and build circuit. Second, using GYC, partial region stability theory is applied to adaptive control for two identical chaotic systems with uncertain parameters. The results of numerical simulation are performed to verify examples of the proposed nonlinear controllers.

Chaos Hybrid Generalized Synchronization of Liu-Chen System by GYC Partial Region Stability Theory

Chaos Hybrid Generalized Synchronization of Liu-Chen System by GYC Partial Region Stability Theory

Adaptive synchronization of chaotic systems with unknown parameters via new backstepping strategy

In this paper, a new effective approach—backstepping with Ge–Yao–Chen (GYC) partial region stability theory (called BGYC in this article) is proposed to applied to adaptive synchronization. Backstepping design is a recursive procedure that combines the choice of a Lyapunov function with the design of a controller, and it presents a systematic procedure for selecting a proper controller in chaos synchronization. We further combine the systematic backstepping design and GYC partial region stability theory in this article, Lyapunov function can be chosen as a simple linear homogeneous function of states, and the controllers and the update laws of parameters shall be much simpler. Further, it also introduces less simulation error—the numerical simulation results show that the states errors and parametric errors approach to zero much more exactly and efficiently, which are compared with the original one. Two cases are presented in the simulation results to show the effectiveness and feasibility of our new strategy.

Fuzzy adaptive synchronization of time-reversed chaotic systems via a new adaptive control strategy

A novel adaptive control strategy is proposed herein to increase the efficiency of adaptive control by combining Takagi–Sugeno (T–S) fuzzy modeling and the Ge–Yao–Chen (GYC) partial region stability theory. This approach provides two major contributions: (1) increased synchronization efficiency, especially for parameters tracing and (2) a simpler controller design. Two simulated cases are presented for comparison: Case 1 utilizes normal adaptive synchronization, whereas Case 2 utilizes the Takagi–Sugeno (T–S) fuzzy model-based Lorenz systems to realize adaptive synchronization via the new adaptive scheme. The simulation results demonstrate the effectiveness and feasibility of our new adaptive strategy.

The Chaos Generalized Synchronization of a Quantum-CNN Chaotic Oscillator with a Double Duffing Chaotic System by GYC Partial Region Stability Theory

The Chaos Generalized Synchronization of a Quantum-CNN Chaotic Oscillator with a Double Duffing Chaotic System by GYC Partial Region Stability Theory

Chaos generalized synchronization of an inertial tachometer with new Mathieu-Van der Pol systems as functional system by GYC partial region stability theory

A new strategy to achieve generalized chaos synchronization by GYC partial region stability theory is proposed. By using the GYC partial region stability theory the Lyapunov function is a simple linear homogeneous function of error states and the controllers are more simple and introduce less simulation error because they are in lower order than that of traditional controllers. In simulation examples, an inertial tachometer system and Mathieu-Van der Pol system are used.

Chaos generalized synchronization of new Mathieu–Van der Pol systems with new Duffing–Van der Pol systems as functional system by GYC partial region stability theory

In this paper, a new strategy by using GYC partial region stability theory is proposed to achieve generalized chaos synchronization. via using the GYC partial region stability theory, the new Lyapunov function used is a simple linear homogeneous function of states and the lower order controllers are much more simple and introduce less simulation error. Numerical simulations are given for new Mathieu–Van der Pol system and new Duffing–Van der Pol system to show the effectiveness of this strategy.

A novel study of parity and attractor in the time reversed Lorentz system

In this Letter, a new effective approach to achieve adaptive synchronization is proposed. Via using Ge–Yao–Chen (GYC) partial region stability theory and pragmatical asymptotically stability theorem, the numerical simulation results show that the states errors and parameter errors approach to zero much more exactly and efficiently than traditional method. The time reversed Lorenz system (called historical Lorenz system in this Letter) is introduced and used for example in this Letter. The simulation results are given in figures and tables for comparison between the new approach and traditional one to show the effectiveness and feasibility of our new strategy.

Chaos control of new Mathieu–Van der Pol systems with new Mathieu–Duffing systems as functional system by GYC partial region stability theory

In this paper, a new strategy by using GYC partial region stability theory is proposed to achieve chaos control. Using the GYC partial region stability theory, the new Lyapunov function used is a simple linear homogeneous function of error states and the lower order controllers are much more simple and introduce less simulation error. Numerical simulations are given for new Mathieu–Van der Pol system and new Mathieu–Duffing system to show the effectiveness of this strategy.

Chaos control of new Ikeda–Lorenz systems by GYC partial region stability theory

AbstractA new strategy to achieve chaos control by GYC partial region stability theory is proposed. By using the GYC partial region stability theory, the Lyapunov function is a simple linear homogeneous function of error states, the controllers are more simple and have less simulation error because they are in lower degree than that of traditional controllers. Simulation results for a new Ikeda–Lorenz system show the effectiveness of this strategy. Copyright © 2008 John Wiley & Sons, Ltd.

Chaos control and anticontrol of a tachometer system by Ge—Yao—Chen partial region stability theory

In this paper, chaos control and anticontrol of a tachometer system by Ge—Yao—Chen (GYC) partial region stability are proposed. The Lyapunov function becomes a simple linear homogeneous function and the controllers become simpler by using the GYC partial region stability theory. The simulation results are more precise because the controllers are in lower degree than that of traditional controllers. Finally, chaos control and anticontrol of the tachometer system by GYC partial region stability are obtained and verified by numerical simulations.