Fractional-order Lorenz System Research Articles

Linear matrix inequality criteria for robust synchronization of uncertain fractional-order chaotic systems

This paper is devoted to synchronization of uncertain fractional-order chaotic systems with fractional-order α: 0 < α < 1 and 1 ≤ α < 2, respectively. On the basis of the stability theory of fractional-order differential system and the observer-based robust control, two sufficient and necessary conditions for synchronizing uncertain fractional-order chaotic systems with parameter perturbations are presented in terms of linear matrix inequality, which is an efficient method and could be easily solved by the toolbox of MATLAB. Finally, fractional-order uncertain chaotic Lü system with fractional-order α = 0.95 and fractional-order uncertain chaotic Lorenz system with fractional-order α = 1.05 are taken as numerical examples to show the validity and feasibility of the proposed method.

Design of sliding mode controller for a class of fractional-order chaotic systems

In this paper, a sliding mode control law is designed to control chaos in a class of fractional-order chaotic systems. A class of unknown fractional-order systems is introduced. Based on the sliding mode control method, the states of the fractional-order system have been stabled, even if the system with uncertainty is in the presence of external disturbance. In addition, chaos control is implemented in the fractional-order Chen system, the fractional-order Lorenz system, and the same to the fractional-order financial system by utilizing this method. Effectiveness of the proposed control scheme is illustrated through numerical simulations.

Lag projective synchronization in fractional-order chaotic (hyperchaotic) systems

In this Letter, a new lag projective synchronization for fractional-order chaotic (hyperchaotic) systems is proposed, which includes complete synchronization, anti-synchronization, lag synchronization, generalized projective synchronization. It is shown that the slave system synchronizes the past state of the driver up to a scaling factor. A suitable controller for achieving the lag projective synchronization is designed based on the stability theory of linear fractional-order systems and the pole placement technique. Two examples are given to illustrate effectiveness of the scheme, in which the lag projective synchronizations between fractional-order chaotic Rössler system and fractional-order chaotic Lü system, between fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system, respectively, are successfully achieved. Corresponding numerical simulations are also given to verify the analytical results.

Synchronization of Coupled Nonidentical Fractional-Order Hyperchaotic Systems

Synchronization of coupled nonidentical fractional-order hyperchaotic systems is addressed by the active sliding mode method. By designing an active sliding mode controller and choosing proper control parameters, the master and slave systems are synchronized. Furthermore, synchronizing fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system is performed to show the effectiveness of the proposed controller.

Can We Obtain a Fractional Lorenz System from a Physical Problem?

A new fractional-order Lorenz system is obtained from the convection of fractional Maxwell fluids in a circular loop. This is the first fractional-order dynamical system derived from an actual physical problem, and rich dynamical properties are observed. In the case of short fluid relaxation time, with the decreasing effective dimension Σ, we find a critical value of the effective dimension Σcr1, at which the solution of the system undergoes a transition from the chaotic motion to the periodic motion and another critical value Σcr2(Σcr2 < Σcr1) at which the regular dynamics of the system returns to the chaotic one. In the case of long relaxation time, the phenomenon of overstability is observed and the decrease of Σ is found to delay the onset of it.

Controlling fractional order chaotic systems based on Takagi–Sugeno fuzzy model and adaptive adjustment mechanism

In this Letter, a kind of novel model, called the generalized Takagi–Sugeno (T-S) fuzzy model, is first developed by extending the conventional T-S fuzzy model. Then, a simple but efficient method to control fractional order chaotic systems is proposed using the generalized T-S fuzzy model and adaptive adjustment mechanism (AAM). Sufficient conditions are derived to guarantee chaos control from the stability criterion of linear fractional order systems. The proposed approach offers a systematic design procedure for stabilizing a large class of fractional order chaotic systems from the literature about chaos research. The effectiveness of the approach is tested on fractional order Rössler system and fractional order Lorenz system.

Function projective synchronization for fractional-order chaotic systems

This letter investigates the function projective synchronization between fractional-order chaotic systems. Based on the stability theory of fractional-order systems and tracking control, a controller for the synchronization of two fractional-order chaotic systems is designed. This technique is applied to achieve synchronization between the fractional-order Lorenz systems with different orders, and achieve synchronization between the fractional-order Lorenz system and fractional-order Chen system. The numerical simulations demonstrate the validity and feasibility of the proposed method.

BIFURCATIONS AND CHAOS IN FRACTIONAL-ORDER SIMPLIFIED LORENZ SYSTEM

The dynamics of fractional-order systems have attracted increasing attention in recent years. In this paper, we numerically study the bifurcations and chaotic behaviors in the fractional-order simplified Lorenz system using the time-domain scheme. Chaos does exist in this system for a wide range of fractional orders, both less than and greater than three. Complex dynamics with interesting characteristics are presented by means of phase portraits, bifurcation diagrams and the largest Lyapunov exponent. Both the system parameter and the fractional order can be taken as bifurcation parameters, and the range of existing chaos is different for different parameters. The lowest order we found for this system to yield chaos is 2.62.

Circuit design and implementation of Lorenz chaotic system with one parameter

To study the characteristics of the chaotic systems and their applications, an electronic circuit of simplified Lorenz chaotic system with one parameter is designed and experimented with discrete components. The system parameters correspond to the circuit element parameters. By regulating the variable resistor in the circuit, dynamic behaviors including limit cycle, pitchfork bifurcation, period-doubling bifurcation, chaos, and route to chaos by period-doubling bifurcation, are observed. The necessary condition for the existence of chaos in the fractional-order simplified Lorenz system is deduced. The lowest order of the fractional-order simplified Lorenz system and the variation law of the lowest order with system parameters are determined. Circuit simulations and experiments show that the simplified Lorenz system has rich dynamic characteristics, and that theoretical analysis and circuit experiment are accordant with each other.

Synchronization of fractional-order unified chaotic system via linear control

Chaos synchronization in fractional-order unified chaotic system is disscussed in this paper. Based on the stability theory of fractional-order system, the control law is presented to achieve chaos synchronization. The advantage of the proposed controllers is that they are linear and have lower dimensions than that of the states. With this technique it is very easy to find the suitable feedback constant. Simulation results for fractional-order Lorenz, Lü and Chen chaotic systems are provided to illustrate the effectiveness of the proposed scheme.

CHAOTIC SYNCHRONIZATION OF FRACTIONAL-ORDER MODIFIED COUPLED DYNAMOS SYSTEM

This paper studies the problem of chaotic synchronization of the fractional-order modified coupled dynamos system, designs the activate controllers, and then proves that the self-synchronization of the fractional-order modified coupled dynamos system and the fractional-order modified coupled dynamos system's different structure synchronization with the fractional-order Lorenz system can both arrive theoretically. The simulation results demonstrate the validity of the activate controller.

Chaos in the fractional-order Lorenz system

In this article, we investigate the chaotic behaviours in the fractional-order Lorenz system. By utilizing the fractional calculus techniques, we found that chaos exists in the fractional-order Lorenz system of order less than 3. The lowest order we found to have chaos in this system is 2.97.

Stability Analysis of a Class of Nonlinear Fractional-Order Systems

In this paper, a stability theorem of nonlinear fractional-order differential equations is proven theoretically by using the Gronwall-Bellman lemma. According to this theorem, the linear state feedback controller is introduced for stabilizing a class of nonlinear fractional-order systems. And, a new criterion is derived for designing the controller gains for stabilization, in which control parameters can be selected via the pole placement technique of the linear fractional-order control theory. Finally, the theoretical results are further substantiated by simulation results of the fractional-order chaotic Lorenz system with desired design requirements.

The synchronization of three chaotic fractional-order Lorenz systems with bidirectional coupling

This paper mainly investigates the chaos synchronization of three identical fractional order Lorenz systems with linearly coupling. And the sufficient conditions to realize synchronization are obtained by Laplace transform theory. Numerical solutions and simulations are used for demonstration.

Chaos in fractional-order Liu system and a fractional-order unified system and the synchronization between them

This paper studies the chaotic behaviors of the fractional-order Liu system and the fractional-order unified system which were presented several years ago. It is found that chaos exists in the both systems with order less than three and the lowest order to have chaos is 0.3. The calculation results of the maximum Lyapunov exponents when the system is chaotic illustrate the existence of chaos. Chaos synchronization between fractional-order Liu system, fractional-order Lorenz system and fractional-order Lü sytem is realized by employing active control technique. Theoretieal analysis and numerical simulations demonstrate the effectiveness of the proposed method.