Stabilization of fractional-order chaotic system via a single state adaptive-feedback controller

This letter investigates the stabilization of three-dimensional fractional-order chaotic systems, and proposes a single state adaptive-feedback controller for fractional-order chaos control based on Lyapunov stability theory, fractional order differential inequality, and adaptive control theory. The present controller which only contains a single state variable is simple both in design and implementation. Simulation results for several fractional-order chaotic systems are provided to illustrate the effectiveness of the proposed scheme.     

Adaptive synchronization of fractional-order chaotic systems via a single driving variable

This letter investigates the synchronization of a class of three-dimensional fractional-order chaotic systems. Based on sliding mode variable structure control theory and adaptive control technique, a single-state adaptive-feedback controller containing a novel fractional integral sliding surface is developed to synchronize a class of fractional-order chaotic systems. The present controller, which only contains a single driving variable, is simple both in design and implementation. Simulation results for three fractional-order chaotic systems are provided to illustrate the effectiveness of the proposed scheme.     

Circuit simulation for synchronization of a fractional-order and integer-order chaotic system

We design a new three-dimensional double-wing fractional-order chaotic system with three quadratic terms, confirmed by numerical simulation and circuit implementation. We then study the synchronization between the new double-wing fractional-order chaotic system and different Lorenz systems with different structures. In the process of the synchronization, the definition of ‘the simplest response system’ and the practical method of designing the circuit have been originally proposed. The circuit of ‘the simplest response system’ (even the simplest incommensurate-order response system), holding different structures with the drive system, of any one integral or fractional drive system now can be designed effectively and sufficiently. Our results are supported by numerical simulation and circuit implementation.     

New class of chaotic systems with equilibrium points like a three-leaved clover

This paper presents a new class of chaotic systems with infinite number of equilibrium points like a three-leaved clover. They signify an exciting class of dynamical systems which represent many major characteristics of regular and chaotic motions. These chaotic systems belong to the general class of chaotic systems with hidden attractors. By using a systematic computer search, three chaotic systems with three-leaved-clover-shaped equilibria were found which are classified into dissipative systems. Dynamics of the chaotic system with the three-leaved-clover-equilibria has been investigated by using phase portraits, bifurcation diagram, Lyapunov exponents, Kaplan–Yorke dimension and Poincaré map. Moreover, an electronic circuit implementation of the theoretical system is designed to check its effectiveness. Random number generator design has been realized with newly developed chaotic systems. The obtained random bit sequences are used for image encryption. Security analysis of image encryption processes has been performed.     

Enhanced FPGA realization of the fractional-order derivative and application to a variable-order chaotic system

Nonlinear Dynamics - The efficiency of the hardware implementations of fractional-order systems heavily relies on the efficiency of realizing the fractional-order derivative operator. In this work,...     

Response to the comments on “Adaptive synchronization of fractional-order chaotic systems via a single driving variable”

This paper is a Response to the comment by M.P. Aghababa (Nonlinear Dyn. 2001, doi:). Some ideas are proposed to rebut the Comments.     

Synchronization between integer-order chaotic systems and a class of fractional-order chaotic system based on fuzzy sliding mode control

In this paper, we focus on the synchronization between integer-order chaotic systems and a class of fractional-order chaotic system using the stability theory of fractional-order systems. A new fuzzy sliding mode method is proposed to accomplish this end for different initial conditions and number of dimensions. Furthermore, three examples are presented to illustrate the effectiveness of the proposed scheme, which are the synchronization between a fractional-order Lü chaotic system and an integer-order Liu chaotic system, the synchronization between a fractional-order hyperchaotic system based on Chen??s system and an integer-order hyperchaotic system based upon the Lorenz system, and the synchronization between a fractional-order hyperchaotic system based on Chen??s system, and an integer-order Liu chaotic system. Finally, numerical results are presented and are in agreement with theoretical analysis.     

Stabilization of the motion of an unstable linear mechanical system

Kiev. Translated from Prikladnaya Mekhanika, Vol. 26, No. 4, pp. 79–85, April, 1990.     

Stabilization of the Furuta Pendulum Based on a Lyapunov Function

We propose a Lyapunov-function-based control for the stabilization of the under-actuated Furuta pendulum. Firstly, by a suitable partial feedback linearization that allows to linearize only the actuated coordinate of the system, we proceed to find a candidate Lyapunov function. Based on this candidate function, we derive a stabilizing controller, in such away that the closed-loop system is locally and asymptotically stable around the unstable vertical equilibrium rest, with a computable domain of attraction.     

Lyapunov spectrum of chaotic maps with a long-range coupling mediated by a diffusing substance

We investigate analytically and numerically coupled lattices of chaotic maps where the interaction is non-local, i.e., each site is coupled to all the other sites but the interaction strength decreases exponentially with the lattice distance. This kind of coupling models an assembly of pointlike chaotic oscillators in which the coupling is mediated by a rapidly diffusing chemical substance. We consider a case of a lattice of Bernoulli maps, for which the Lyapunov spectrum can be analytically computed and also the completely synchronized state of chaotic Ulam maps, for which we derive analytically the Lyapunov spectrum.     

Stabilizing the unstable periodic orbits of a chaotic system using model independent adaptive time-delayed controller

In this paper we deal with the control of chaotic systems. Knowing that a chaotic attractor contains a myriad of unstable periodic orbits (UPO’s), the aim of our work is to stabilize some of the UPO’s embedded in the chaotic attractor and which have interesting characteristics. First, using the input-to-state linearization method in conjunction with a time-delayed state feedback, we design a control signal that can achieve stabilization. Next, an adaptive time-delayed state feedback is proposed which shows at once efficiency and simplicity and circumvents the construction complexity of the first controller. Finally, we propose a reduced order sliding mode observer to estimate the necessary states for the design of an adaptive time delayed state feedback controller. This last controller has one main advantage, it in fact achieves UPO stabilization without using the system model. The efficacy of the proposed methods is illustrated by numerical simulations onto Chua’s system.     

Stabilization of the unstable steady-state regimes of a flow reactor

Stabilization of the unstable regime of a flow reactor with distributed parameters is considered. The stabilizing system relates the rate of supply of material to the reactor to the deviation of the output concentration from the steady-state value. An isothermal autocatalytic reaction is chosen as the simplest reaction for which nonuniqueness of the steady-state regimes is possible. It has been found [3] that there are three possible steady-state regimes. Their stability was investigated in [8]. It has been shown that the middle regime is unstable, only the first mode of the small disturbances being unstable over a broad range of the parameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 115–119, March–April, 1989.     

Lyapunov spectrum determination from the FEM simulation of a chaotic advecting flow

The problem of the determination of the Lyapunov spectrum in chaotic advection using approximated velocity fields resulting from a standard FEM method is investigated. A fourth order Runge–Kutta scheme for trajectory integration is combined with a third order Jacobian matrix method with QR ‐factorization. After checking the algorithm on the standard Lorenz and coupled quartic oscillator systems, the method is applied to a model 3‐D steady flow for which an analytical expression is known. Both linear and quadratic approximated velocity fields succeed in predicting the Lyapunov exponents as well as describing the chaotic or regular regions inside the flow with satisfactory accuracy. A more realistic flow is then studied in order to delineate the possible limitations of the approach. Copyright © 2005 John Wiley & Sons, Ltd.     

Bifurcation points of the elastic-plastic equilibrium in a simple frame model

Summary This is a study of the characteristics of equilibrium bifurcation in a geometrically symmetric simple frame model in which the beam is loaded symmetrically and gradually. The deformability of the model is concentrated in two Shanley cells for whose elements a nonholonomic bilinear force-lengthening (or force-contraction) relation is adopted.The equilibrium bifurcation characteristics of this model are compared with those of Shanley's uniformly stressed column model. The characteristics of the two models coincide when instantaneous plasticisation of the deformable elements for the frame model are assumed, as occurs in the column model. But with a succession of different plastic states as the load increases, differences emerge. The most important being: in the frame model the borderline between the interval of unstable bifurcation points and the interval of stable bifurcation points is not necessarily, as it is in the column model, marked by a neutral (stationary load) bifurcation point.

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Sommario S'indagano i caratteri della biforcazione dell'equilibrio di un modello di portale, simmetrico nella geometria e simmetricamente caricato sul traverso in modo graduale. La deformabilità del modello è concentrata in due celle alla Shanley, per i cui elementi si adotta una relazione bilineare anolonoma tra forze ed allungamenti (od accorciamenti).Si confrontano tali caratteri con quelli della biforcazione dell'equilibrio del ben noto modello di asta caricata di punta studiato da Shanley.Quando si ipotizzi, per il modello di portale, una plasticizzazione istantanea degli elementi deformabili (proprio come avviene nel modello di asta), si ha perfetta coincidenza dei caratteri in esame.All'opposto, quando si ipotizzi un succedersi di diversi stati plastici al crescere del carico, si constatano delle differenze; in particolare, per il modello di portale esiste la possibilità, a differenza di quanto avviene nel modello di asta, che non sia un punto di biforcazione neutra (a carico stazionario) a segnare il confine tra l'intervallo dei punti di biforcazione instabile e quello dei punti di biforcazione stabile.

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Study supported by the Plasticity Group of the National Research Council.     

Synchronization of chaotic system with uncertain variable parameters by linear coupling and pragmatical adaptive tracking

We study the synchronization of general chaotic systems which satisfy the Lipschitz condition only, with uncertain variable parameters by linear coupling and pragmatical adaptive tracking. The uncertain parameters of a system vary with time due to aging, environment, and disturbances. A?sufficient condition is given for the asymptotical stability of common zero solution of error dynamics and parameter update dynamics by the Ge?CYu?CChen pragmatical asymptotical stability theorem based on equal probability assumption. Numerical results are studied for a Lorenz system and a quantum cellular neural network oscillator to show the effectiveness of the proposed synchronization strategy.     

External stability of Caputo fractional-order nonlinear control systems: advances in the Lyapunov function method

Nonlinear Dynamics - Nonlinear behavior of bubbles, and most importantly $$\frac{1}{2}$$ -order subharmonics (SH), are used to increase the contrast-to-tissue ratio (CTR) in diagnostic ultrasound...