Fractional-order Differential Systems Research Articles

Maximum and minimum solutions for a nonlocal p-Laplacian fractional differential system from eco-economical processes

This paper focuses on the maximum and minimum solutions for a fractional order differential system, involving a p-Laplacian operator and nonlocal boundary conditions, which arises from many complex processes such as ecological economy phenomena and diffusive interaction. By introducing new type growth conditions and using the monotone iterative technique, some new results about the existence of maximal and minimal solutions for a fractional order differential system is established, and the estimation of the lower and upper bounds of the maximum and minimum solutions is also derived. In addition, the iterative schemes starting from some explicit initial values and converging to the exact maximum and minimum solutions are also constructed.

Pinning synchronization of fractional-order complex networks by a single controller

We investigate the state feedback pinning synchronization of fractional-order complex networks. Based on the stability theory of fractional-order differential systems and state feedback control by a single controller, synchronization conditions for fractional-order complex networks are given. We assume that the coupling matrix is irreducible, and provide a numerical example to illustrate the validity of the proposed conclusions.

The Fractional-Order mathematical modeling of bacterial resistance against multiple antibiotics in case of local bacterial infection

In this study, it is described the general forms of fractional-order differential equations and asymtotic stability of their system’s equilibria. In addition that, the stability analysis of equilibrium points of the local bacterial infection model which is fractional-order differential equation system, is made. Results of this analysis are supported via numerical simulations drawn by datas obtained from literature for mycobacterium tuberculosis and the antibiotics isoniazid (INH), rifampicin (RIF), streptomycin (SRT) and pyrazinamide (PRZ) used against this bacterial infection.

PINNING SYNCHRONIZATION OF FRACTIONAL-ORDER COMPLEX NETWORKS BY A SINGLE CONTROLLER

We investigate the state feedback pinning synchronization of fractional-order complex networks. Based on the stability theory of fractional-order differential systems and state feedback control by a single controller, synchronization conditions for fractional-order complex networks are given. We assume that the coupling matrix is irreducible, and provide a numerical example to illustrate the validity of the proposed conclusions.

Global projective synchronization in finite time of nonidentical fractional-order neural networks based on sliding mode control strategy

In this paper, a novel fractional-order sliding mode control strategy is introduced to synchronize two nonidentical fractional order neural networks (FNNs) in finite time. Firstly, by applying fractional-order calculation, the properties of the global asymptotical stability and convergence in finite time are developed for the fractional-order differential equation system. Secondly, based on the proposed principle of convergence in finite time, a new fractional-order sliding mode surface is presented and its global stability in finite time is analytically proved. In addition, an appropriate sliding mode controller is designed to drive the state trajectories of error system to the prescribed sliding surface in finite time and remain on it evermore. Meanwhile, by means of the fractional Lyapunov-like approach, the global projective synchronization condition is given, and the synchronization time is explicitly evaluated. As the special case, some sufficient conditions are given to guarantee global complete synchronization, anti-synchronization and stabilization of nonidentical FNNs. Finally, an example is given to demonstrate the validity of the proposed method.

Stability by linear approximation and the relation between the stability of difference and differential fractional systems

The paper is devoted to study the stability of nonlinear fractional order difference systems by their linear approximation. Additionally, we show the relation between the stability of linear fractional order differential systems and their discretizations. Copyright © 2016 John Wiley & Sons, Ltd.

Positive solutions of singular fractional order differential system with Riemann-Stieltjes integral boundary condition

In this paper, we study the existence of positive solutions to a class of higher-order nonlinear fractional functional differential system with Riemann-Stieltjes integral boundary conditions. Our method relies upon the upper and lower solutions and the Schauder fixed point theorem. Furthermore, we constructed an iterative scheme to approximate the positive solution. We also give an example to illustrate the main results.

Adaptive feedback synchronization of fractional-order complex dynamic networks

The aim of this paper is to investigate synchronization of fractional-order complex dynamic networks. To ensure synchronization of two dynamic networks, an adaptive feedback control method is proposed. With the stability analysis of the fractional-order differential system, we rigorously prove that the controller can make trajectory errors between the drive and response networks synchronized. The simple but practical method can be applied to a class of fractional-order networks without any prior analytical knowledge of the systems. Three illustrative examples are given to show the effectiveness of the proposed method.

Parameter estimation and topology identification of uncertain general fractional-order complex dynamical networks with time delay

Complex networks have attracted much attention from various fields of sciences and engineering in recent years. However, many complex networks have various uncertain information, such as unknown or uncertain system parameters and topological structure, which greatly affects the system dynamics. Thus, the parameter estimation and structure identification problem has theoretical and practical importance for uncertain complex dynamical networks. This paper investigates identification of unknown system parameters and network topologies in uncertain fractional-order complex network with time delays (including coupling delay and node delay). Based on the stability theorem of fractional-order differential system and the adaptive control technique, a novel and general method is proposed to address this challenge. Finally two representative examples are given to verify the effectiveness of the proposed approach.

A Class of Integer Order and Fractional Order Hyperchaotic Systems via the Chen System

In this article, we investigate the generation of a class of hyperchaotic systems via the Chen chaotic system using both integer order and fractional order differential equation systems. Based on the Chen chaotic system, we designed a system with four nonlinear ordinary differential equations. For different parameter sets, the trajectory of the system may diverge or display a hyperchaotic attractor with double wings. By linearizing the ordinary differential equation system with divergent trajectory and designing proper switching controls, we obtain a chaotic attractor. Similar phenomenon has also been observed in linearizing the hyperchaotic system. The corresponding fractional order systems are also considered. Our investigation indicates that, switching control can be applied to either linearized chaotic or nonchaotic differential equation systems to create chaotic attractor.

Dynamical analysis, stabilization and discretization of a chaotic fractional-order GLV model

In this paper, the fractional-order generalized Lotka–Volterra (GLV) model and its discretization are investigated qualitatively. A sufficient condition for existence and uniqueness of the solution of the proposed system is shown. Analytical conditions of the stability of the system’s three non-negative steady states are proved. The conditions of the existence of Hopf bifurcation in the fractional-order GLV system are discussed. The necessary conditions for this system to remain chaotic are obtained. Based on the stability theory of fractional-order differential systems, a new control scheme is introduced to stabilize the fractional-order GLV system to its steady states. Furthermore, the analytical conditions of stability of the discretized system are also studied. It is shown that the system’s fractional parameter has effect on the stability of the discretized system which shows rich variety of dynamical analysis such as bifurcations, an attractor crisis and chaotic attractors. Numerical simulations are used to support the analytical results.

An efficient approach to design confusion creating component and its analyses using majority logic criterion

In 1949, the concept of confusion creating component was proposed by some cryptographers for the sake of good security, so, from then to now, every block cipher encryption system has this component as an essential part. The aim of this work is to propose a scheme for the assembly of 8×8 nonlinear confusion component for encryption algorithms based on 'fractional-order hyper chaotic Lorentz differential equation system' (FOHCLS). The security analyses of the proposed confusion component and its utility in image encryption application show that it is invulnerable against renowned attacks.

Synchronization of fractional-order chaotic systems using unidirectional adaptive full-state linear error feedback coupling

Based on the stability theory of fractional-order system, a novel unidirectional adaptive full-state linear error feedback coupling scheme is extended to control and synchronize all of fractional-order differential (FOD) chaotic systems with in-commensurate (and commensurate) orders. The feedback strength is adaptive to an updated law rather than prescribed as a constant. The convergence speed of feedback strength is regulated by a constant. With rigorous linear algebraic theorems and precisely numerical matrix computations, a reasonable interval in which the ultimate final control strength dwells is suggested, and the reliability of synchronization state is guaranteed. It demonstrates that the unidirectional full-state linear feedback coupling scheme can be adopted to control and synchronize FOD chaotic systems directly. Numerical simulations of three representative FOD chaotic systems illustrate the effectiveness of the proposed scheme.

Leader-following consensus of fractional-order multi-agent systems via adaptive pinning control

In this paper, the leader-following consensus problem of fractional-order multi-agent systems is considered via adaptive pinning control. The dynamics of leader and all followers with linear and nonlinear functions are investigated, respectively. We assume that the node should be pinned if its in-degree is less than its out-degree in the paper. Under this assumption and based on the stability theory of fractional-order differential systems, some leader-following consensus criteria are derived, which are easily obtained by matrix inequalities. The control of each agent using local information is designed and detailed analysis of the leader-following consensus is presented. The design technique is based on algebraic graph theory and the Riccati inequality. Several simulation examples are presented to demonstrate the effectiveness of the proposed method.

Adaptive hybrid projective synchronization of two coupled fractional-order complex networks with different sizes

This paper investigates a new hybrid projective synchronization scheme between two coupled fractional-order complex networks with different sizes. The hybrid projective synchronization studied in this paper includes complete synchronization of the states of the nodes in each network and projective synchronization of the states of a pair of nodes from both networks. Based on the stability theorem of fractional-order differential system and adaptive control technique, some sufficient conditions for guaranteeing the existence of the hybrid projective synchronization are derived. Two examples are given to show the effectiveness of the proposed methods.

Razumikhin-type stability theorems for functional fractional-order differential systems and applications

In this paper, we studied the stability of functional fractional-order differential systems using a Lyapunov function and develop Razumikhin-type uniform stability and global uniform asymptotic stability theorems for functional fractional-order differential systems, which involve Riemann–Liouville and Caputo derivatives respectively. Two examples are given as application of our theorems.

A Series of New Chaotic Attractors via Switched Linear Integer Order and Fractional Order Differential Equations

This article investigates the design of a series of new chaotic attractors. A switching control with different switching surfaces is designed to link two systems of linear integer order differential equations. Under such control, the linked systems have rich dynamical behaviors such as chaos. We also investigate the dynamical behaviors of the corresponding linear fractional order differential equation systems with switching controls. It is shown that such fractional order systems have chaotic behaviors as well.

Periodic BVP for a class of nonlinear differential equation with a deviated argument and integrable impulses

This paper deals with periodic BVP for integer/fractional order differential equations with a deviated argument and integrable impulses in arbitrary Banach space X for which the impulses are not instantaneous. By utilizing fixed point theorems, we firstly establish the existence and uniqueness of the mild solution for the integer order differential system and secondly obtain the existence results for the mild solution to the fractional order differential system. Also at the end, we present some examples to show the effectiveness of the discussed abstract theory.

The Generation of a Series of Multiwing Chaotic Attractors Using Integer and Fractional Order Differential Equation Systems

In this article, we present a systematic approach to design chaos generators using integer order and fractional order differential equation systems. A series of multiwing chaotic attractors and grid multiwing chaotic attractors are obtained using linear integer order differential equation systems with switching controls. The existence of chaotic attractors in the corresponding fractional order differential equation systems is also investigated. We show that, using the nonlinear fractional order differential equation system, or linear fractional order differential equation systems with switching controls, a series of multiwing chaotic attractors can be obtained.

On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives

In this paper, a class of nonlinear fractional order differential impulsive systems with Hadamard derivative is discussed. First, a reasonable concept on the solutions of fractional impulsive Cauchy problems with Hadamard derivative and the corresponding fractional integral equations are established. Second, two fundamental existence results are presented by using standard fixed point methods. Finally, two examples are given to illustrate our theoretical results.