A note on the multi-stage spectral relaxation method for chaos control and synchronization

In this study, we present and apply a new, accurate and easy to implement numerical method to realize and verify the synchronization between two identical chaotic Lorenz, Genesio-Tesi, Rössler, Chen and Rikitake systems. The proposed method is called the multi-stage spectral relaxation method (MSRM). We utilize the active control technique for the synchronization of these systems. To illustrate the effectiveness of the method, simulation results are presented and compared with results obtained using the Runge-Kutta (4, 5) based MATLAB solver, ode45

A note on the multi-stage spectral relaxation method for chaos control and synchronization

In this study, we present and apply a new, accurate and easy to implement numerical method to realize and verify the synchronization between two identical chaotic Lorenz, Genesio-Tesi, Rössler, Chen and Rikitake systems. The proposed method is called the multi-stage spectral relaxation method (MSRM). We utilize the active control technique for the synchronization of these systems. To illustrate the effectiveness of the method, simulation results are presented and compared with results obtained using the Runge-Kutta (4, 5) based MATLAB solver, ode45

Pseudo-State Sliding Mode Control of Fractional SISO Nonlinear Systems

This paper deals with the problem of pseudo-state sliding mode control of fractional SISO nonlinear systems with model inaccuracies. Firstly, a stable fractional sliding mode surface is constructed based on the Routh-Hurwitz conditions for fractional differential equations. Secondly, a sliding mode control law is designed using the theory of Mittag-Leffler stability. Further, we utilize the control methodology to synchronize two fractional chaotic systems, which serves as an example of verifying the viability and effectiveness of the proposed technique

Transcritical and zero-Hopf bifurcations in the Genesio system

Agraïments: The first author is supported by FAPESP Grant No. 2013/24541-0. Both authors are supported by CAPES Grant 88881.030454/2013-01 Program CSF-PVE.In this paper we study the existence of transcritical and zero--Hopf bifurcations of the third--order ordinary differential equation a b c x - x^2 = 0, called the Genesio equation, which has a unique quadratic nonlinear term and three real parameters. More precisely, writing this differential equation as a first order differential system in \R^

Nonlinear Unknown‐Input Observer‐Based Systems for Secure Communication

Secure communication employing chaotic systems is considered in this chapter. Chaos‐based communication uses chaotic systems as its backbone for information transmission and extraction, and is a field of active research and development and rapid advances in the literature. The theory and methods of synchronizing chaotic systems employing unknown input observers (UIOs) are investigated. New and novel results are presented. The techniques developed can be applied to a wide class of chaotic systems. Applications to the estimation of a variety of information signals, such as speech signal, electrocardiogram, stock price data hidden in chaotic system dynamics, are presented

Reduction of dimension for nonlinear dynamical systems

We consider reduction of dimension for nonlinear dynamical systems. We demonstrate that in some cases, one can reduce a nonlinear system of equations into a single equation for one of the state variables, and this can be useful for computing the solution when using a variety of analytical approaches. In the case where this reduction is possible, we employ differential elimination to obtain the reduced system. While analytical, the approach is algorithmic, and is implemented in symbolic software such as {\sc MAPLE} or {\sc SageMath}. In other cases, the reduction cannot be performed strictly in terms of differential operators, and one obtains integro-differential operators, which may still be useful. In either case, one can use the reduced equation to both approximate solutions for the state variables and perform chaos diagnostics more efficiently than could be done for the original higher-dimensional system, as well as to construct Lyapunov functions which help in the large-time study of the state variables. A number of chaotic and hyperchaotic dynamical systems are used as examples in order to motivate the approach.Comment: 16 pages, no figure

Synchronization of two fractional-order chaotic systems via nonsingular terminal Fuzzy Sliding Mode Control

La sincronización de dos sistemas caóticos complejos de orden fraccional se discute en este documento. El parámetro incertidumbre y la perturbación externa se incluyen en el modelo del sistema, y la sincronización de los sistemas considerados caóticos se implementa en base al concepto de tiempo finito. Primero, se propone una nueva superficie deslizante terminal no singular de orden fraccional que es adecuada para los sistemas de orden fraccional considerados. Se ha demostrado que una vez que las trayectorias del estado del sistema alcancen la superficie de deslizamiento propuesta, convergerán al origen dentro de un tiempo finito dado. En segundo lugar, en términos de la superficie deslizante terminal no singular establecida, que combina el control difuso y los esquemas de control de modo deslizante, se introduce una ley de control de modo deslizante difusa única novedosa y robusta que puede forzar las trayectorias del sistema de error dinámico de circuito cerrado para alcanzar el deslizamiento superficie durante un tiempo finito. Finalmente, utilizando el teorema de estabilidad fraccional de Lyapunov, se demuestra la estabilidad del método propuesto. El método propuesto se implementa para la sincronización de dos sistemas caóticos Genesio-Tesi de orden fraccional con parámetros inciertos y perturbaciones externas para verificar la efectividad del controlador de modo de deslizamiento difuso terminal no-singular de orden fraccional propuesto.The synchronization of two fractional-order complex chaotic systems is discussed in this paper. The parameter uncertainty and external disturbance are included in the system model, and the synchronization of the considered chaotic systems is implemented based on the finite-time concept. First, a novel fractional-order nonsingular terminal sliding surface which is suitable for the considered fractional-order systems is proposed. It is proven that once the state trajectories of the system reach the proposed sliding surface they will converge to the origin within a given finite time. Second, in terms of the established nonsingular terminal sliding surface, combining the fuzzy control and the sliding mode control schemes, a novel robust single fuzzy sliding mode control law is introduced, which can force the closed-loop dynamic error system trajectories to reach the sliding surface over a finite time. Finally, using the fractional Lyapunov stability theorem, the stability of the proposed method is proven. The proposed method is implemented for synchronization of two fractional-order Genesio-Tesi chaotic systems with uncertain parameters and external disturbances to verify the effectiveness of the proposed fractional-order nonsingular terminal fuzzy sliding mode controller.• National Natural Science Foundation of China. Becas U1604146, U1404610, 61473115, 61203047 • Science and Technology Research Project in Henan Province. 152102210273, 162102410024 • Foundation for the University Technological Innovative Talents of Henan Province. Beca 18HASTIT019peerReviewe