FPGA implementation of novel fractional-order chaotic systems with two equilibriums and no equilibrium and its adaptive sliding mode synchronization

This paper introduces two novel fractional-order chaotic systems with cubic nonlinear resistor and investigates its adaptive sliding mode synchronization. Firstly the novel two equilibrium chaotic system with cubic nonlinear resistor (NCCNR) is derived and its dynamic properties are investigated. The fractional-order cubic nonlinear resistor system (FONCCNR) is then derived from the integer-order model and its stability and fractional-order bifurcation are discussed. Next a novel no-equilibrium chaotic cubic nonlinear resistor system (NECNR) is derived from NCCNR system. Dynamic properties of NECNR system are investigated. The fractional-order no equilibrium cubic nonlinear resistor system (FONECNR) is derived from NECNR. Stability and fractional-order bifurcation are investigated for the FONECNR system. The non-identical adaptive sliding mode synchronization of FONCCNR and FONECNR systems are achieved. Finally the proposed systems, adaptive control laws, sliding surfaces and adaptive controllers are implemented in FPGA.     

A novel non-equilibrium fractional-order chaotic system and its complete synchronization by circuit implementation

In this paper, we construct a novel four dimensional fractional-order chaotic system. Compared with all the proposed chaotic systems until now, the biggest difference and most attractive place is that there exists no equilibrium point in this system. Those rigorous approaches, i.e., Melnikov??s and Shilnikov??s methods, fail to mathematically prove the existence of chaos in this kind of system under some parameters. To reconcile this awkward situation, we resort to circuit simulation experiment to accomplish this task. Before this, we use improved version of the Adams?CBashforth?CMoulton numerical algorithm to calculate this fractional-order chaotic system and show that the proposed fractional-order system with the order as low as 3.28 exhibits a chaotic attractor. Then an electronic circuit is designed for order q=0.9, from which we can observe that chaotic attractor does exist in this fractional-order system. Furthermore, based on the final value theorem of the Laplace transformation, synchronization of two novel fractional-order chaotic systems with the help of one-way coupling method is realized for order q=0.9. An electronic circuit is designed for hardware implementation to synchronize two novel fractional-order chaotic systems for the same order. The results for numerical simulations and circuit experiments are in very good agreement with each other, thus proving that chaos exists indeed in the proposed fractional-order system and the one-way coupling synchronization method is very effective to this system.     

Using fractional-order integrator to control chaos in single-input chaotic systems

This paper deals with a fractional calculus based control strategy for chaos suppression in the 3D chaotic systems. It is assumed that the structure of the controlled chaotic system has only one control input. In the proposed strategy, the controller has three tuneable parameters and the control input is constructed as fractional-order integration of a linear combination of linearized model states. The tuning procedure is based on the stability theorems in the incommensurate fractional-order systems. To evaluate the performance of the proposed controller, the design method is applied to suppress chaotic oscillations in a 3D chaotic oscillator and in the Chen chaotic system.     

Synchronization of nonidentical chaotic fractional-order systems with different orders of fractional derivatives

This paper addresses the problem of synchronization of chaotic fractional-order systems with different orders of fractional derivatives. Based on the stability theory of fractional-order linear systems and the idea of tracking control, suitable controllers are correspondingly proposed for two cases: the first is synchronization between two identical chaotic fractional-order systems with different fractional orders, and the other is synchronization between two nonidentical fractional-order chaotic systems with different fractional orders. Three numerical examples illustrate that fast synchronization can be achieved even between a chaotic fractional-order system and a hyperchaotic fractional-order system.     

Chaos detection and parameter identification in fractional-order chaotic systems with delay

The paper first applies the 0–1 test for chaos to detecting chaos exhibited by fractional-order delayed systems. The results of the test reveal that there exists chaos in some fractional-order delayed systems with specific parameter values, which coincides with previous reports based on the phase portrait. In addition, it is very important to identify exactly the unknown specific parameters of fractional-order chaotic delayed systems in chaos control and synchronization. Thus, a method for parameter identification of fractional-order chaotic delayed systems based on particle swarm optimization (PSO) is presented. By treating the orders as parameters, the parameters and orders are identified through minimizing an objective function. PSO can efficiently find the optimal feasible solution of the objective function. Finally, numerical simulations on fractional-order chaotic logistic delayed system and fractional-order chaotic Chen delayed system show that the proposed method has effective performance of parameter identification.     

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.     

Chaos in the fractional-order complex Lorenz system and its synchronization

In this article, a novel dynamic system, the fractional-order complex Lorenz system, is proposed. Dynamic behaviors of a fractional-order chaotic system in complex space are investigated for the first time. Chaotic regions and periodic windows are explored as well as different types of motion shown along the routes to chaos. Numerical experiments by means of phase portraits, bifurcation diagrams and the largest Lyapunov exponent are involved. A new method to search the lowest order of the fractional-order system is discussed. Based on the above result, a synchronization scheme in fractional-order complex Lorenz systems is presented and the corresponding numerical simulations demonstrate the effectiveness and feasibility of the proposed scheme.     

Robust synchronization of two different fractional-order chaotic systems with unknown parameters using adaptive sliding mode approach

In this paper, an adaptive sliding mode control method is introduced to ensure robust synchronization of two different fractional-order chaotic systems with fully unknown parameters and external disturbances. For this purpose, a fractional integral sliding surface is defined and an adaptive sliding mode controller is designed. In this method, no knowledge of the bounds of parameters and perturbation is required in advance and the parameters are updated through an adaptive control process. The proposed scheme is global and theoretically rigorous. Two examples are given to illustrate effectiveness of the scheme, in which the synchronizations between fractional-order chaotic Chen system and fractional-order chaotic Rössler 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.

     

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.     

Finite-time chaos control and synchronization of fractional-order nonautonomous chaotic (hyperchaotic) systems using fractional nonsingular terminal sliding mode technique

In this paper, a novel fractional-order terminal sliding mode control approach is introduced to control/synchronize chaos of fractional-order nonautonomous chaotic/hyperchaotic systems in a given finite time. The effects of model uncertainties and external disturbances are fully taken into account. First, a novel fractional nonsingular terminal sliding surface is proposed and its finite-time convergence to zero is analytically proved. Then an appropriate robust fractional sliding mode control law is proposed to ensure the occurrence of the sliding motion in a given finite time. The fractional version of the Lyapunov stability is used to prove the finite-time existence of the sliding motion. The proposed control scheme is applied to control/synchronize chaos of autonomous/nonautonomous fractional-order chaotic/hyperchaotic systems in the presence of both model uncertainties and external disturbances. Two illustrative examples are presented to show the efficiency and applicability of the proposed finite-time control strategy. It is worth to notice that the proposed fractional nonsingular terminal sliding mode control approach can be applied to control a broad range of nonlinear autonomous/nonautonomous fractional-order dynamical systems in finite time.     

Robust chaos synchronization of fractional-order chaotic systems with unknown parameters and uncertain perturbations

Chaotic systems in practice are always influenced by some uncertainties and external disturbances. This paper investigates the problem of practical synchronization of fractional-order chaotic systems. Based on Lyapunov stability theory and a fractional-order differential inequality, a modified adaptive control scheme and adaptive laws of parameters are developed to robustly synchronize coupled fractional-order chaotic systems with unknown parameters and uncertain perturbations. This synchronization approach is simple, global and theoretically rigorous. Simulation results for two fractional-order chaotic systems are provided to illustrate the effectiveness of the proposed scheme.     

A practical synchronization approach for fractional-order chaotic systems

A practical synchronization approach is proposed for a class of fractional-order chaotic systems to realize perfect \(\delta \)-synchronization, and the nonlinear functions in the fractional-order chaotic systems are all polynomials. The \(\delta \)-synchronization scheme in this paper means that the origin in synchronization error system is stable. The reliability of \(\delta \)-synchronization has been confirmed on a class of fractional-order chaotic systems with detailed theoretical proof and discussion. Furthermore, the \(\delta \)-synchronization scheme for the fractional-order Lorenz chaotic system and the fractional-order Chua circuit is presented to demonstrate the effectiveness of the proposed method.     

Lyapunov-based fractional-order controller design to synchronize a class of fractional-order chaotic systems

In this paper, a novel adaptive fractional-order feedback controller is first developed by extending an adaptive integer-order feedback controller. Then a simple but practical method to synchronize almost all familiar fractional-order chaotic systems has been put forward. Through rigorous theoretical proof by means of the Lyapunov stability theorem and Barbalat lemma, sufficient conditions are derived to guarantee chaos synchronization. A wide range of fractional-order chaotic systems, including the commensurate system and incommensurate case, autonomous system, and nonautonomous case, is just the novelty of this technique. The feasibility and validity of presented scheme have been illustrated by numerical simulations of the fractional-order Chen system, fractional-order hyperchaotic Lü system, and fractional-order Duffing system.     

Robust synchronization of uncertain fractional-order chaotic systems with time-varying delay

This paper presents a new technique using a recurrent non-singleton type-2 sequential fuzzy neural network (RNT2SFNN) for synchronization of the fractional-order chaotic systems with time-varying delay and uncertain dynamics. The consequent parameters of the proposed RNT2SFNN are learned based on the Lyapunov–Krasovskii stability analysis. The proposed control method is used to synchronize two non-identical and identical fractional-order chaotic systems, with time-varying delay. Also, to demonstrate the performance of the proposed control method, in the other practical applications, the proposed controller is applied to synchronize the master–slave bilateral teleoperation problem with time-varying delay. Simulation results show that the proposed control scenario results in good performance in the presence of external disturbance, unknown functions in the dynamics of the system and also time-varying delay in the control signal and the dynamics of system. Finally, the effectiveness of proposed RNT2SFNN is verified by a nonlinear identification problem and its performance is compared with other well-known neural networks.     

A new chaotic system with fractional order and its projective synchronization

Based on Rikitake system, a new chaotic system is discussed. Some basic dynamical properties, such as equilibrium points, Lyapunov exponents, fractal dimension, Poincaré map, bifurcation diagrams and chaotic dynamical behaviors of the new chaotic system are studied, either numerically or analytically. The obtained results show clearly that the system discussed is a new chaotic system. By utilizing the fractional calculus theory and computer simulations, it is found that chaos exists in the new fractional-order three-dimensional system with order less than 3. The lowest order to yield chaos in this system is 2.733. The results are validated by the existence of one positive Lyapunov exponent and some phase diagrams. Further, based on the stability theory of the fractional-order system, projective synchronization of the new fractional-order chaotic system through designing the suitable nonlinear controller is investigated. The proposed method is rather simple and need not compute the conditional Lyapunov exponents. Numerical results are performed to verify the effectiveness of the presented synchronization scheme.     

Hopf bifurcation and chaos in fractional-order modified hybrid optical system

In this paper, a chaotic fractional-order modified hybrid optical system is presented. Some basic dynamical properties are further investigated by means of Poincaré mapping, parameter phase portraits, and the largest Lyapunov exponents. Fractional Hopf bifurcation conditions are proposed; it is found that Hopf bifurcation occurs on the proposed system when the fractional-order varies and passes a sequence of critical values. The chaotic motion is validated by the positive Lyapunov exponent. Finally, some numerical simulations are also carried out to illustrate our results.     

No-chatter variable structure control for fractional nonlinear complex systems

This paper concerns the problem of robust control of uncertain fractional-order nonlinear complex systems. After establishing a simple linear sliding surface, the sliding mode theory is used to derive a novel robust fractional control law for ensuring the existence of the sliding motion in finite time. We use a nonsmooth positive definitive function to prove the stability of the controlled system based on the fractional version of the Lyapunov stability theorem. In order to avoid the chattering, which is inherent in conventional sliding mode controllers, we transfer the sign function of the control input into the first derivative of the control signal. The proposed sliding mode approach is also applied for control of a class of nonlinear fractional-order systems via a single control input. Simulation results indicate that the proposed fractional variable structure controller works well for stabilization of hyperchaotic and chaotic complex fractional-order nonlinear systems. Moreover, it is revealed that the control inputs are free of chattering and practical.     

Dynamics analysis and hybrid function projective synchronization of a new chaotic system

This paper introduces a novel three-dimensional autonomous chaotic system by adding a quadratic cross-product term to the first equation and modifying the state variable in the third equation of a chaotic system proposed by Cai et al. (Acta Phys. Sin. 56:6230, 2007). By means of theoretical analysis and computer simulations, some basic dynamical properties, such as Lyapunov exponent spectrum, bifurcations, equilibria, and chaotic dynamical behaviors of the new chaotic system are investigated. Furthermore, hybrid function projective synchronization (HFPS) of the new chaotic system is studied by employing three different synchronization methods, i.e., adaptive control, system coupling and active control. The proposed approaches are applied to achieve HFPS between two identical new chaotic systems with fully uncertain parameters, HFPS in coupled new chaotic systems, and HFPS between the integer-order new chaotic system and the fractional-order Lü chaotic system, respectively. Corresponding numerical simulations are provided to validate and illustrate the analytical results.     

A novel fractional-order hyperchaotic system stabilization via fractional sliding-mode control

In this paper we numerically investigate the fractional-order sliding-mode control for a novel fractional-order hyperchaotic system. Firstly, the dynamic analysis approaches of the hyperchaotic system involving phase portraits, Lyapunov exponents, bifurcation diagram, Lyapunov dimension, and Poincaré maps are investigated. Then the fractional-order generalizations of the chaotic and hyperchaotic systems are studied briefly. The minimum orders we found for chaos and hyperchaos to exist in such systems are 2.89 and 3.66, respectively. Finally, the fractional-order sliding-mode controller is designed to control the fractional-order hyperchaotic system. Numerical experimental examples are shown to verify the theoretical results.     

Modified projective synchronization of fractional-order chaotic systems via active sliding mode control

This paper is devoted to study the problem of modified projective synchronization of fractional-order chaotic system. Base on the stability theorems of fractional-order linear system, active sliding mode controller is proposed to synchronize two different fractional-order systems. Moreover, the controller is robust to the bounded noise. Numerical simulations are provided to show the effectiveness of the analytical results.