Numerical approach to chaotic pattern formation in diffusive predator–prey system with Caputo fractional operator

This paper is primarily concern with the formulation and analysis of a reliable numerical method based on the novel alternating direction implicit finite difference scheme for the solution of the fractional reaction–diffusion system. In the work, the integer first‐order derivative in time is replaced with the Caputo fractional derivative operator. As a case study, the dynamics of predator–prey model is considered. In order to provide a good guidelines on the correct choice of parameters for the numerical simulation of full fractional reaction–diffusion system, its linear stability analysis is also examined. The resulting scheme is applied to solve both self‐diffusion and cross‐diffusion problems in two‐dimensions. We observed in the experimental results a range of spatiotemporal and chaotic structures that are related to Turing pattern. It was also discovered in the simulations that cross‐diffusive case gives rise to spatial patterns faster than the diffusive case. Apart from chaotic spiral‐like structures obtained in this work, it should also be mentioned that Turing patterns such as stationary spots and stripes are obtainable, depending on the initial and parameters choices.     

Switching adaptive controllers to control fractional‐order complex systems with unknown structure and input nonlinearities

This article investigates the chaos control problem for the fractional‐order chaotic systems containing unknown structure and input nonlinearities. Two types of nonlinearity in the control input are considered. In the first case, a general continuous nonlinearity input is supposed in the controller, and in the second case, the unknown dead‐zone input is included. In each case, a proper switching adaptive controller is introduced to stabilize the fractional‐order chaotic system in the presence of unknown parameters and uncertainties. The control methods are designed based on the boundedness property of the chaotic system's states, where, in the proposed methods the nonlinear/linear dynamic terms of the fractional‐order chaotic systems are assumed to be fully unknown. The analytical results of the mentioned techniques are proved by the stability analysis theorem of fractional‐order systems and the adaptive control method. In addition, as an application of the proposed methods, single input adaptive controllers are adopted for control of a class of three‐dimensional nonlinear fractional‐order chaotic systems. And finally, some numerical examples illustrate the correctness of the analytical results. © 2014 Wiley Periodicals, Inc. Complexity 21: 211–223, 2015     

Exponential synchronization for fractional‐order chaotic systems with mixed uncertainties

This article focuses on the problem of exponential synchronization for fractional‐order chaotic systems via a nonfragile controller. A criterion for α‐exponential stability of an error system is obtained using the drive‐response synchronization concept together with the Lyapunov stability theory and linear matrix inequalities approach. The uncertainty in system is considered with polytopic form together with structured form. The sufficient conditions are derived for two kinds of structured uncertainty, namely, (1) norm bounded one and (2) linear fractional transformation one. Finally, numerical examples are presented by taking the fractional‐order chaotic Lorenz system and fractional‐order chaotic Newton–Leipnik system to illustrate the applicability of the obtained theory. © 2014 Wiley Periodicals, Inc. Complexity 21: 114–125, 2015     

Chaos control and synchronization of fractional order delay‐varying computer virus propagation model

In the present article, the authors have studied the dynamical behavior of delay‐varying computer virus propagation (CVP) model with fractional order derivative, and it is found that the chaotic attractor exists in the considered fractional order system. In order to eliminate the chaotic behavior of fractional order delay‐varying CVP model, feedback controlmethod is used. This article also dealswith the synchronization between controlled and chaotic delay‐varying CVPmodel via active controlmethod. The fractional derivative is described in the Caputo sense. Numerical simulation results are carried out by means of Adams‐Boshforth‐Moultonmethod with the help ofMATLAB, and the results are successfully depicted through graphs .Copyright © 2015 John Wiley & Sons, Ltd.     

Fractional modeling and control of a complex nonlinear energy supply‐demand system

This article deals with the fractional‐order modeling of a complex four‐dimensional energy supply‐demand system (FOESDS). First, the fractional calculus techniques are adopted to describe the dynamics of the energy supply‐demand system. Then the complex behavior of the proposed fractional‐order FOESDS is studied using numerical simulations. It is shown that the FOESDS can exhibit stable, chaotic, and unstable states. When it exhibits chaos, the FOESDS's strange attractors are plotted to validate the chaotic behavior of the system. Moreover, we calculate the maximal Lyapunov exponents of the system to confirm the existence of chaos. Accordingly, to stabilize the system, a finite‐time active fractional‐order controller is proposed. The effects of model uncertainties and external disturbances are also taken into account. An estimation of the stabilization time is given. Based on the latest version of the fractional Lyapunov stability theory, the finite‐time stability and robustness of the proposed method are proved. Finally, two illustrative examples are provided to illustrate the usefulness and applicability of the proposed control scheme. © 2014 Wiley Periodicals, Inc. Complexity 20: 74–86, 2015     

Multi‐switching combination–combination synchronization of non‐identical fractional‐order chaotic systems

In this paper, multi‐switching combination–combination synchronization scheme has been investigated between a class of four non‐identical fractional‐order chaotic systems. The fractional‐order Lorenz and Chen's systems are taken as drive systems. The combination–combination of multi drive systems is then synchronized with the combination of fractional‐order Lü and Rössler chaotic systems. In multi‐switching combination–combination synchronization, the state variables of two drive systems synchronize with different state variables of two response systems simultaneously. Based on the stability of fractional‐order chaotic systems, the multi‐switching combination–combination synchronization of four fractional‐order non‐identical systems has been investigated. For the synchronization of four non‐identical fractional‐order chaotic systems, suitable controllers have been designed. Theoretical analysis and numerical results are presented to demonstrate the validity and feasibility of the applied method. Copyright © 2017 John Wiley & Sons, Ltd.     

Projective synchronization between different fractional‐order hyperchaotic systems with uncertain parameters using proposed modified adaptive projective synchronization technique

In the present article, the authors have proposed a modified projective adaptive synchronization technique for fractional‐order chaotic systems. The adaptive projective synchronization controller and identification parameters law are developed on the basis of Lyapunov direct stability theory. The proposed method is successfully applied for the projective synchronization between fractional‐order hyperchaotic Lü system as drive system and fractional‐order hyperchaotic Lorenz chaotic system as response system. A comparison between the effects on synchronization time due to the presence of fractional‐order time derivatives for modified projective synchronization method and proposed modified adaptive projective synchronization technique is the key feature of the present article. Numerical simulation results, which are carried out using Adams–Boshforth–Moulton method show that the proposed technique is effective, convenient and also faster for projective synchronization of fractional‐order nonlinear dynamical systems. Copyright © 2013 John Wiley & Sons, Ltd.     

Synchronization and antisynchronization of N‐coupled fractional‐order complex chaotic systems with ring connection

This paper is devoted to investigate synchronization and antisynchronization of N‐coupled general fractional‐order complex chaotic systems described by a unified mathematical expression with ring connection. By means of the direct design method, the appropriate controllers are designed to transform the fractional‐order error dynamical system into a nonlinear system with antisymmetric structure. Thus, by using the recently established result for the Caputo fractional derivative of a quadratic function and a fractional‐order extension of the Lyapunov direct method, several stability criteria are derived to ensure the occurrence of synchronization and antisynchronization among N‐coupled fractional‐order complex chaotic systems. Moreover, numerical simulations are performed to illustrate the effectiveness of the proposed design.     

Chaotic fractional‐order model for muscular blood vessel and its control via fractional control scheme

This article studies the chaotic and complex behavior in a fractional‐order biomathematical model of a muscular blood vessel (MBV). It is shown that the fractional‐order MBV (FOMBV) model exhibits very complex and rich dynamics such as chaos. We show that the corresponding maximal Lyapunov exponent of the FOMBV system is positive which implies the existence of chaos. Strange attractors of the FOMBV model are depicted to validate the chaotic behavior of the system. We change the fractional order of the model and investigate the dynamics of the system. To suppress the chaotic behavior of the model, we propose a single input fractional finite‐time controller and prove its stability using the fractional Lyapunov theory. In addition, the effects of the model uncertainties and external disturbances are taken into account and a robust fractional finite‐time controller is constructed. The upper bound of the chaos suppression time is also given. Some computer simulations are presented to illustrate the findings of this article. © 2014 Wiley Periodicals, Inc. Complexity 20: 37–46, 2014     

Generation of multi-wing chaotic attractor in fractional order system

In this paper, a novel approach is proposed for generating multi-wing chaotic attractors from the fractional linear differential system via nonlinear state feedback controller equipped with a duality-symmetric multi-segment quadratic function. The main idea is to design a proper nonlinear state feedback controller by using four construction criterions from a fundamental fractional differential nominal linear system, so that the controlled fractional differential system can generate multi-wing chaotic attractors. It is the first time in the literature to report the multi-wing chaotic attractors from an uncoupled fractional differential system. Furthermore, some basic dynamical analysis and numerical simulations are also given, confirming the effectiveness of the proposed method.     

Synchronization of fractional order chaotic systems using active control method

In this article, the active control method is used for synchronization of two different pairs of fractional order systems with Lotka–Volterra chaotic system as the master system and the other two fractional order chaotic systems, viz., Newton–Leipnik and Lorenz systems as slave systems separately. The fractional derivative is described in Caputo sense. Numerical simulation results which are carried out using Adams–Bashforth–Moulton method show that the method is easy to implement and reliable for synchronizing the two nonlinear fractional order chaotic systems while it also allows both the systems to remain in chaotic states. A salient feature of this analysis is the revelation that the time for synchronization increases when the system-pair approaches the integer order from fractional order for Lotka–Volterra and Newton–Leipnik systems while it reduces for the other concerned pair.     

Synchronization for fractional‐order multi‐linked complex network with two kinds of topological structure via periodically intermittent control

This paper concentrates on the global synchronization of the fractional‐order multi‐linked complex network (FMCN) via periodically intermittent control. It should be stressed that periodically intermittent control is employed to the FMCN for the first time. Moreover, the network is defined on digraphs with different weights, and two situations on topological structure of the network are discussed, including each digraph being strongly connected, and the biggest one being strongly connected. Based on Lyapunov method and graph theory, some synchronization criteria are obtained under two situations. And, the obtained synchronization criteria have a close relationship with the order of fractional‐order derivative, coupling strength, control gain, control rate, and control period. Besides, for practicability, theoretical results are applied to studying the synchronization of fractional‐order multi‐linked chaotic systems, and some sufficient conditions are provided. For a special case, fractional‐order multi‐linked Lorenz chaotic systems, numerical simulations are given to indicate the feasibility of theoretical results and the effectiveness of control strategy.     

Adaptive synchronization of new fractional‐order chaotic systems with fractional adaption laws based on risk analysis

In this paper, a new fractional‐order chaotic system and an adaptive synchronization of fractional‐order chaotic system are proposed. Parameters adaption laws are obtained to design adaptive controllers using Lyapunov stability theory of fractional‐order system. Finally, reliability of designed controllers and risk analysis of adaptive synchronization problem are formulated and, risk of using the proposed controllers in presences of external disturbances are demonstrated. Also, risk of controllers are reduced using an optimizing method. Numerical examples are used to verify the performance of the proposed controllers.     

Synchronization of fractional‐order chaotic systems with disturbances via novel fractional‐integer integral sliding mode control and application to neuron models

In this paper, a novel fractional‐integer integral type sliding mode technique for control and generalized function projective synchronization of different fractional‐order chaotic systems with different dimensions in the presence of disturbances is presented. When the upper bounds of the disturbances are known, a sliding mode control rule is proposed to insure the existence of the sliding motion in finite time. Furthermore, an adaptive sliding mode control is designed when the upper bounds of the disturbances are unknown. The stability analysis of sliding mode surface is given using the Lyapunov stability theory. Finally, the results performed for synchronization of three‐dimensional fractional‐order chaotic Hindmarsh‐Rose (HR) neuron model and two‐dimensional fractional‐order chaotic FitzHugh‐Nagumo (FHN) neuron model.     

Limitations of frequency domain approximation for detecting chaos in fractional order systems

In this paper, we analytically study the influences of using frequency domain approximation in numerical simulations of fractional order systems. The number and location of equilibria, and also the stability of these points, are compared between the original system and its frequency based approximated counterpart. It is shown that the original system and its approximation are not necessarily equivalent according to the number, location and stability of the fixed points. This problem can cause erroneous results in special cases. For instance, to prove the existence of chaos in fractional order systems, numerical simulations have been largely based on frequency domain approximations, but in this paper we show that this method is not always reliable for detecting chaos. This approximation can numerically demonstrate chaos in the non-chaotic fractional order systems, or eliminate chaotic behavior from a chaotic fractional order system.     

A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter

In recent years chaotic secure communication and chaos synchronization have received ever increasing attention. In this paper, for the first time, a fractional chaotic communication method using an extended fractional Kalman filter is presented. The chaotic synchronization is implemented by the EFKF design in the presence of channel additive noise and processing noise. Encoding chaotic communication achieves a satisfactory, typical secure communication scheme. In the proposed system, security is enhanced based on spreading the signal in frequency and encrypting it in time domain. In this paper, the main advantages of using fractional order systems, increasing nonlinearity and spreading the power spectrum are highlighted. To illustrate the effectiveness of the proposed scheme, a numerical example based on the fractional Lorenz dynamical system is presented and the results are compared to the integer Lorenz system.     

Numerical solutions of nonlinear fractional Schrödinger equations using nonstandard discretizations

In this article, numerical study for both nonlinear space‐fractional Schrödinger equation and the coupled nonlinear space‐fractional Schrödinger system is presented. We offer here the weighted average nonstandard finite difference method (WANSFDM) as a novel numerical technique to study such kinds of partial differential equations. The space fractional derivative is described in the sense of the quantum Riesz‐Feller definition. Stability analysis of the proposed method is studied. To show that this method is reliable and computationally efficient different numerical examples are provided. We expect that the proposed schemes can be applicable to different systems of fractional partial differential equations. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1399–1419, 2017     

分数阶Duffing混沌系统的动力性态及其由单一主动控制的混沌同步

随着物理与技术的深入研究,分数阶非线性系统的动力性态及其分数阶混沌系统的同步成为研究的焦点.研究了分数阶Duffing系统的动力性态包括混沌性质,并且由分数阶非线性稳定性准则得到了分数阶非自治系统的混沌同步.特别地,研究了由单一主动控制的分数阶Duffing系统的同步.相应的数值结果演示了方法的有效性.     

Solution and dynamics analysis of a fractional‐order hyperchaotic system

Numerical solution and chaotic behaviors of the fractional‐order simplified Lorenz hyperchaotic system are investigated in this paper. The solution of the fractional‐order hyperchaotic system is obtained by employing Adomian decomposition method. Lyapunov characteristic exponents algorithm for the fractional‐order chaotic system is designed. Dynamics of the fractional‐order hyperchaotic system are analyzed by means of bifurcation diagrams, Lyapunov characteristic exponents, C₀ complexity, and chaos diagram. It shows that this system has rich dynamical behaviors, and it is more complex when the fractional order q is small. It lays a foundation for the practical application of the fractional‐order hyperchaotic systems. Copyright © 2015 John Wiley & Sons, Ltd.     

Chaos control of fractional order Rabinovich–Fabrikant system and synchronization between chaotic and chaos controlled fractional order Rabinovich–Fabrikant system

In this article the local stability of the Rabinovich–Fabrikant (R–F) chaotic system with fractional order time derivative is analyzed using fractional Routh–Hurwitz stability criterion. Feedback control method is used to control chaos in the considered fractional order system and after controlling the chaos the authors have introduced the synchronization between fractional order non-chaotic R–F system and the chaotic R–F system at various equilibrium points. The fractional derivative is described in the Caputo sense. Numerical simulation results which are carried out using Adams–Boshforth–Moulton method show that the method is effective and reliable for synchronizing the systems.