Adaptive feedback control and synchronization of non-identical chaotic fractional order systems

This paper addresses the reliable synchronization problem between two non-identical chaotic fractional order systems. In this work, we present an adaptive feedback control scheme for the synchronization of two coupled chaotic fractional order systems with different fractional orders. Based on the stability results of linear fractional order systems and Laplace transform theory, using the master-slave synchronization scheme, sufficient conditions for chaos synchronization are derived. The designed controller ensures that fractional order chaotic oscillators that have non-identical fractional orders can be synchronized with suitable feedback controller applied to the response system. Numerical simulations are performed to assess the performance of the proposed adaptive controller in synchronizing chaotic systems.     

Synchronization of a novel fractional order stretch-twist-fold (STF) flow chaotic system and its application to a new authenticated encryption scheme (AES)

In this paper, a new fractional order stretch-twist-fold (STF) flow dynamical system is proposed. The stability analysis of the proposed system equilibria is accomplished and we establish that the system is exhibited chaos even for order less than 3. The active control method is applied to enquire the hybrid phase synchronization between two identical fractional order STF flow chaotic systems. These synchronized systems are applied to formulate an authenticated encryption scheme newly for message (text and image) recovery. It is widely applied in the field of secure communication. Numerical simulations are presented to validate the effectiveness of the proposed theory.     

Dynamic behavior of fractional order Duffing chaotic system and its synchronization via singly active control

With the increasingly deep studies in physics and technology,the dynamics of fractional order nonlinear systems and the synchronization of fractional order chaotic systems have become the focus in scientific research.In this paper,the dynamic behavior including the chaotic properties of fractional order Duffing systems is extensively investigated.With the stability criterion of linear fractional systems,the synchronization of a fractional non-autonomous system is obtained.Specifically,an effective singly active control is proposed and used to synchronize a fractional order Duffing system.The numerical results demonstrate the effectiveness of the proposed methods.     

Fractional Rayleigh–Duffing-like system and its synchronization

This paper presents a periodically driven Rayleigh?CDuffing-like system with the function x|x|. It is proven via the Melnikov function method that the quadratic function x|x| induces Smale horseshoes to the Rayleigh?CDuffing-like system. The Rayleigh?CDuffing-like oscillator with fractional order is also discussed, and results of computer simulation demonstrate the chaotic dynamic behaviors of the system. Furthermore, two fractional Rayleigh?CDuffing-like systems are synchronized by active control technology, the method based on state observer and nonlinear feedback method. Numerical results validate the effectiveness and applicability of the proposed synchronization schemes.     

Fractional order fixed-time nonsingular terminal sliding mode synchronization and control of fractional order chaotic systems

This paper presents fractional order fixed-time nonsingular terminal sliding mode control for stabilization and synchronization of fractional order chaotic systems with uncertainties and disturbances. First, a novel fractional order terminal sliding mode surface is proposed to guarantee the fixed-time convergence of system states along the sliding surface. Second, a nonsingular terminal sliding mode controller is designed to force the system states to reach the sliding surface within fixed-time and remain on it forever. Furthermore, the fractional Lyapunov stability theory is used to prove the fixed-time stability and the robustness of the proposed control scheme and estimate the upper bound of convergence time. Next, the proposed control scheme is applied to the synchronization of two nonidentical fractional order Liu chaotic systems and chaos suppression of fractional order power system. Simulation results verify the effectiveness of the proposed control scheme. Finally, some application issues about the proposed scheme are discussed.

     

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 suppression in a fractional order financial system using intelligent regrouping PSO based fractional fuzzy control policy in the presence of fractional Gaussian noise

Financial systems are known to have irregular and erratic fluctuations due to diverse influences and often result in economic crisis and huge financial losses. Recent models of financial systems show that they behave chaotically and have long range memory dependence. Mitigating these undesirable chaotic natures of financial systems by appropriate control policies is important in order to reduce investment risks and improve economic performance. In this paper, a fractional order fuzzy control policy is employed to suppress the chaotic dynamics of a representative chaotic fractional order financial system. An intelligent Regrouping Particle Swarm Optimization (Reg-PSO) is used to design the numeric weights of the control policy and the methodology is demonstrated by credible simulations. The designed fractional fuzzy control policies are shown to work well with respect to conventional fuzzy control policies in the presence of persistent and anti-persistent noise, which can be due to additional extraneous influences on the system.     

Hybrid robust modified function projective lag synchronization in two different dimensional chaotic systems

In this article, a novel synchronization scheme, modified function projective lag synchronization (MFPLS) in two different dimensional chaotic systems with parameter perturbations, is proposed. In the proposed method, the states of two nonidentical chaotic systems with different orders are asymptotically lag synchronized up to a desired scaling function matrix by means of reduced order and increased order, respectively. Furthermore, based on the reality situation, the parameter perturbations are involved, which are assumed to appear in both drive and response systems. With the Lyapunov stability theory, an adaptive controller is designed to achieve MFPLS. Theoretical proof and numerical simulations demonstrate the effectiveness and feasibility of the proposed scheme.     

Stability criteria for a class of fractional order systems

This paper deals with the stability problem in LTI fractional order systems having fractional orders between 1 and 1.5. Some sufficient algebraic conditions to guarantee the BIBO stability in such systems are obtained. The obtained conditions directly depend on the coefficients of the system equations, and consequently using them is easier than the use of conditions constructed based on solving the characteristic equation of the system. Some illustrations are presented to show the applicability of the obtained conditions. For example, it is shown that these conditions may be useful in stabilization of unstable fractional order systems or in taming fractional order chaotic systems.     

Application of Takagi–Sugeno fuzzy model to a class of chaotic synchronization and anti-synchronization

In this study, we investigate a class of chaotic synchronization and anti-synchronization with stochastic parameters. A controller is composed of a compensation controller and a fuzzy controller which is designed based on fractional stability theory. Three typical examples, including the synchronization between an integer-order Chen system and a fractional-order Lü system, the anti-synchronization of different 4D fractional-order hyperchaotic systems with non-identical orders, and the synchronization between a 3D integer-order chaotic system and a 4D fractional-order hyperchaos system, are presented to illustrate the effectiveness of the controller. The numerical simulation results and theoretical analysis both demonstrate the effectiveness of the proposed approach. Overall, this study presents new insights concerning the concepts of synchronization and anti-synchronization, synchronization and control, the relationship of fractional and integer order nonlinear systems.     

Describing function based methods for predicting chaos in a class of fractional order differential equations

This paper deals with two different methods for predicting chaotic dynamics in fractional order differential equations. These methods, which have been previously proposed for detecting chaos in classical integer order systems, are based on using the describing function method. One of these methods is constructed based on Genesio–Tesi conjecture for existence of chaos, and another method is introduced based on Hirai conjecture about occurrence of chaos in a nonlinear system. These methods are restated to use in predicting chaos in a fractional order differential equation of the order between 2 and 3. Numerical simulation results are presented to show the ability of these methods to detect chaos in two fractional order differential equations with quadratic and cubic nonlinearities.     

A chaotic memcapacitor oscillator with two unstable equilibriums and its fractional form with engineering applications

A novel charge-controlled memcapacitor 3D chaotic oscillator with two unstable equilibriums is proposed. Various dynamic properties of the proposed system are derived and investigated to show the existence of chaotic oscillations. Fractional-order analysis of the chaotic oscillator shows that the maximum value for the largest positive Lyapunov exponent is exhibited in fractional order. Adomian decomposition method is used to discretize the fractional-order system. Field-programmable gate arrays are used to realize the proposed oscillator. In addition, random number generator is designed by employing this novel chaotic system in its fractional-order form.     

Adaptive synchronization of the energy resource systems with mismatched parameters via linear feedback control

Synchronization of energy resource systems with mismatched parameters is investigated. An adaptive linear feedback control scheme for the synchronization of energy resource systems is proposed when the parameters of the master system are unknown and different from those of the slave system. Based on the Lyapunov stability theory, an adaptive control law is derived to make the states of two slightly mismatched chaotic systems asymptotically synchronized. Finally, numerical simulations are performed to verify the proposed results.     

A speech encryption using fractional chaotic systems

In this paper, a new speech encryption using fractional chaotic systems is presented. A two-channel transmission method is used where the original speech is encoded using a nonlinear function of the chaotic states. Conditions for synchronization between fractional chaotic systems through one variable have been investigated theoretically by using the Laplace transform. The keys, key space, key selection rules, and sensitivity to keys are discussed in detail. Results show that the original speech is well masked in the ciphertexts yet recovered faithfully and efficiently by the present schemes.     

Stability analysis of fractional order systems based on T–S fuzzy model with the fractional order \alpha : 0<\alpha <1

This paper addresses the problems of the robust stability and stabilization for fractional order systems based on uncertain Takagi–Sugeno fuzzy model. A sufficient condition of asymptotical stability for fractional order uncertain T–S fuzzy model is given, and a parallel distributed compensating fuzzy controller is designed to asymptotically stabilize the model. The sufficient conditions are formulated in the format of linear matrix inequalities. The fractional order T–S fuzzy model of a chaotic system, which has complex nonlinearity, is developed as a test bed. The effectiveness of the approach is tested on fractional order Rössler system and fractional order uncertain Lorenz system.     

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.     

A new fractional order hyperchaotic Rabinovich system and its dynamical behaviors

In the paper, the dynamical behaviors of a new fractional order hyperchaotic Rabinovich system are investigated, which include its local stability, hyperchaos, chaotic control and synchronization. Firstly, a new fractional order hyperchaotic Rabinovich system with Caputo derivative is proposed. Then, the hyperchaotic attractors of the commensurate and incommensurate fractional order hyperchaotic Rabinovich system are found. After that, four linear feedback controllers are designed to stabilize this fractional order system Finally, by using the active control method the synchronization is studied between the fractional order hyperchaotic and chaos controlled Rabinovich system In addition, the theoretical predictions are confirmed by numerical simulations.     

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.     

Parameters identification for chaotic systems based on a modified Jaya algorithm

In this paper, parameters identification for chaotic systems using a modified Jaya algorithm is considered. Firstly, the objective function is formulated based on the variance rate between the responses acquired from the measurement and calculation. Then, the Jaya algorithm is put forward to solve this nonlinear optimization problem. To enhance the performance of the original Jaya, a one-step K-means clustering mechanism and a new updated equation for the best-so-far solution are introduced. To demonstrate the effectiveness of the suggested method, benchmark functions are firstly employed to conduct optimize. Afterward, numerical simulations, including a jerk circuit chaotic system, a hyper-chaotic system and a synchronized chaotic system are used to verify the present algorithm. The simulation results illustrates that the proposed algorithm for chaotic systems is a promising tool with higher identification accuracy, faster convergence rate, as well as stronger robustness.     

A novel active pinning control for synchronization and anti-synchronization of new uncertain unified chaotic systems

This paper discusses the synchronization and anti-synchronization of new uncertain unified chaotic systems (UUCS). Based on the idea of active control, a novel active Pinning control strategy is presented, which only needs a state of new UUCS. The proposed controller can achieve synchronization between a response system and a drive system, and ensure the synchronized robust stability of new UUCS. Numerical simulations of new UUCS show that the controller can make chaotic systems achieve synchronization or anti-synchronization in a quite short period and both are of good robust stability.