Fractional-order Dynamical Systems Research Articles

The Simulation Technique for Fractional Order Systems with Nonzero Initial Value

The paper mainly concerns about nonzero initial value problems of fractional order dynamic systems. Firstly, to solve the limitation of the existing Simulink block diagram scheme with Riemann-Liouville definition, a modified modular approach is proposed. It is based on the improved piecewise approximation for fractional differential operator and integral operator. The method can solve the problem not only under Riemann-Liouville definition, but also Caputo definition. Considering that the fractional integral operator is supposed to own pure integral property, we construct the strict regular approximation model to improve the performance of the origin scheme. Then the method is simplified with the adoption of the state-space model. The effectiveness and the superiority of proposed schemes are verified by detailed fractional order differential equations.

A fractional order chaotic system with a 3D grid of variable attractors

A novel fractional order dynamical system with a variable double-scroll attractor on a line, lattice and 3D grid is introduced. This system belongs to a class of chaotic systems with adjustable variables but with fractional order. Chaos generation only depends on the value of fractional order. As a result, a chaotic attractor is discovered and propagated in y-line. By introducing two extra control parameters, we also observed that the chaotic attractor varies in x-line, z-line, x−y-lattice, x−z-lattice, y−z-lattice, and 3D-grid. Dynamics of the new system are discovered by using phase portraits, bifurcation diagrams, Lyapunov spectrum, Kaplan–Yorke dimension, dissipative measure. Finally, the proposed fractional order system is designed with analog electronic circuits. Circuit results are in concordance with theoretical findings.

A fractional-order predator–prey model with Beddington–DeAngelis functional response and time-delay

Herein, we propose a fractional-order prey–predator dynamical system with Beddington–DeAngelis type functional response and time-delay. We study the existence of various equilibrium points, and sufficient conditions that ensure the local asymptotic stability of the steady states of such system. The system shows a Hopf-bifurcation which depends on the time-delay. The presence of fractional-order and time-delay in the differential model improves the stability of the solutions and enriches the dynamics of the model. Some numerical examples and simulations are provided to validate the derived theoretical results.

Finite time stability of linear fractional order dynam-ical system with variable delays

In this paper, some sufficient condition ensuring finite time stability are derived for a class of linear fractional order dynamical system with variable delay using generalized Gronwall inequality as well as classical Bellman-Gronwall inequality.

Relationship between integer order systems and fractional order system and its two applications

Existence of periodic solutions and stability of fractional order dynamic systems are two important and difficult issues in fractional order systems U+0028 FOS U+0029 field. In this paper, the relationship between integer order systems U+0028 IOS U+0029 and fractional order systems is discussed. A new proof method based on the above involved relationship for the non existence of periodic solutions of rational fractional order linear time invariant systems is derived. Rational fractional order linear time invariant autonomous system is proved to be equivalent to an integer order linear time invariant non-autonomous system. It is further proved that stability of a fractional order linear time invariant autonomous system is equivalent to the stability of another corresponding integer order linear time invariant autonomous system. The examples and state figures are given to illustrate the effects of conclusion derived.

Stability criteria of arbitrary-order neutral dynamic systems with mixed delays

The stability of a neutral dynamic system is a major study in the class of delay systems. In this effort, we study the stability of fractional-order nonlinear dynamic systems utilizing Lyapunov direct method based on fractional calculus (fractional differential operator and fractional integral operator type Riemann–Liouville operators). We suggest a new non-Lipschitz fractional Lyapunov function. This function is a generalization of the well known Lyapunov functions. We investigate the speed of convergence for special cases. Applications are clarified in the sequel.

An efficient numerical method for the optimal control of fractional-order dynamic systems

This paper aims to investigate an efficient numerical scheme for the optimal control of fractional-order dynamic systems. By using the Grünwald–Letnikov approximation for the fractional derivatives and introducing a new transformation in the calculus of variations, the fractional optimal control problem under consideration is converted into a linear programming problem. Then, the internal model principle is employed in order to extend the new scheme for the fractional dynamic systems affected by the external persistent disturbances. Numerical examples and comparative results verify the validity and applicability of the new technique.

On a system of fuzzy fractional differential inclusions with projection operators

The aim of this paper is to introduce and study a class of fuzzy fractional differential inclusions with projection operators, which captures the desired features of both fuzzy differential inclusions and fractional-order projective dynamical systems within the same framework. The existence of solutions for the open situation is proved by using continuous selection theorem. Moreover, the existence of solutions for the closed situation is showed by employing Lipschitzean selection theorem and fixed point theorem, respectively. Furthermore, an application and a numerical example are also given.

Controllability of Fractional-order Dynamical System with Time Varying-Delays

Controllability of Fractional-order Dynamical System with Time Varying-Delays

Complex‐domain stability criteria for fractional‐order linear dynamical systems

Complex‐domain stability criteria for fractional‐order linear dynamical systems

A Technique for Studying a Class of Fractional-Order Nonlinear Dynamical Systems

In this work, we propose a technique to study nonlinear dynamical systems with fractional-order. The main idea of this technique is to transform the fractional-order dynamical system to the integer one based on Jumarie’s modified Riemann–Liouville sense. Many systems in the interdisciplinary fields could be described by fractional-order nonlinear dynamical systems, such as viscoelastic systems, dielectric polarization, electrode-electrolyte polarization, heat conduction, resistance-capacitance-inductance (RLC) interconnect and electromagnetic waves. To deal with integer order dynamical system it would be much easier in contrast with fractional-order system. Two systems are considered as examples to illustrate the validity and advantages of this technique. We have calculated the Lyapunov exponents of these examples before and after the transformation and obtained the same conclusions. We used the integer version of our example to compute numerically the values of the fractional-order and the system parameters at which chaotic and hyperchaotic behaviors exist.

Sliding mode control for generalized robust synchronization of mismatched fractional order dynamical systems and its application to secure transmission of voice messages

This paper proposes a generalized robust synchronization method for different dimensional fractional order dynamical systems with mismatched fractional derivatives in the presence of function uncertainty and external disturbance by a designing sliding mode controller. Based on the proposed theory of generalized robust synchronization criterion, a novel audio cryptosystem is proposed for sending or sharing voice messages secretly via insecure channel. Numerical examples are given to verify the potency of the proposed theories.

Modeling and administration scheduling of fractional-order pharmacokinetic systems

Fractional-order dynamical systems were recently introduced in the field of pharmacokinetics where they proved powerful tools for modeling the absorption, disposition, distribution and excretion of drugs which are liable to anomalous diffusion, deep tissue trapping and other nonlinear phenomena. In this paper we present several ways to simulate such fractional-order pharmacokinetic models and we evaluate their accuracy and complexity on a fractional-order pharmacokinetic model of Amiodarone, an anti-arrhythmic drug. We then propose an optimal administration scheduling scheme and evaluate it on a population of patients.

On disappearance of chaos in fractional systems

In a seminal paper, Grigorenko and Grigorenko [15], numerically studied fractional order dynamical systems (FODS) of the form Dαixi=fi(x1,x2,x3),0≤αi≤1,(i=1,2,3); and showed the existence of chaos in case of fractional Lorenz system when Σ=α1+α2+α3≤3. Since then voluminous numerical work has been done to explore various FODS, in this regard. It is now an established fact that Σ acts as a chaos controlling parameter.In the present article we take a survey of present literature on chaotic behavior of fractional order dynamical systems. Further we numerically explore fractional Chen, Rössler, Bhalekar–Gejji, Lorenz and Liu systems and observe that chaos always disappear if Σ ≤ 2. The existing examples in the literature along with the systems that we have analyzed lead us to conjecture non-existence of chaos if Σ ≤ 2; which in some sense is a generalization of classical Poincaré–Bendixon theorem for FODS.

On Hopf bifurcation in fractional dynamical systems

Fractional order dynamical systems admit chaotic solutions and the chaos disappears when the fractional order is reduced below a threshold value [1]. Thus the order of the dynamical system acts as a chaos controlling parameter. Hence it is important to study the fractional order dynamical systems and chaos. Study of fractional order dynamical systems is still in its infancy and many aspects are yet to be explored.In pursuance to this in the present paper we prove the existence of fractional Hopf bifurcation in case of fractional version of a chaotic system introduced by Bhalekar and Daftardar-Gejji [2]. We numerically explore the (A, B, α) parameter space and identify the regions in which the system is chaotic. Further we find (global) threshold value of fractional order α below which the chaos in the system disappears regardless of values of system parameters A and B.

Suboptimal control of fractional-order dynamic systems with delay argument

In this paper, an efficient linear programming formulation is proposed for a class of fractional-order optimal control problems with delay argument. By means of the Lagrange multiplier in the calculus of variations and using the formula for fractional integration by parts, the Euler–Lagrange equations are derived in terms of a two-point fractional boundary value problem including an advance term as well as the delay argument. The derived equations are then reduced into a linear programming problem by using a Grünwald–Letnikov approximation for the fractional derivatives and introducing a new transformation in the calculus of variations. The new scheme is also effective for the delay fractional optimal control problems influenced by the external persistent disturbances. Numerical simulations and comparative results verify that the proposed approach is efficient and easy to implement.

A novel cascade encryption algorithm for digital images based on anti-synchronized fractional order dynamical systems

In this paper, an active control technique is employed for anti-synchronization between two identical fractional order reverse butterfly-shaped hyperchaotic systems. We have shown that the convergence rate of anti-synchronization error is very faster by increasing the value of an active controller gain. A new algorithm for image encryption and decryption is introduced and established by anti-synchronized fractional order dynamical systems. Experimental results show that the proposed encryption algorithm has high level security against various attacks. Further, it confirms that the new algorithm is more efficient compared to other existing algorithms.

Обработка сигналов и моделирование динамических систем дробного порядка на основе операционного исчисления аппроксимационного типа

Обработка сигналов и моделирование динамических систем дробного порядка на основе операционного исчисления аппроксимационного типа

T–S fuzzy predictive control for fractional order dynamical systems and its applications

In this paper, the fractional order dynamical systems are precisely reproduced based on T–S fuzzy model and the predictive control method is employed for T–S fuzzy fractional order dynamical systems. The proposed control technique is applied for synchronization of two identical and nonidentical with mismatched fractional derivatives of T–S fuzzy fractional order dynamical systems. Also their corresponding necessary conditions are derived to achieve the synchronization. Numerical simulations are demonstrated to validate the effectiveness of the proposed theoretical results.

A new fractional-order sliding mode controller via a nonlinear disturbance observer for a class of dynamical systems with mismatched disturbances

This paper investigates the stabilization and disturbance rejection for a class of fractional-order nonlinear dynamical systems with mismatched disturbances. To fulfill this purpose a new fractional-order sliding mode control (FOSMC) based on a nonlinear disturbance observer is proposed. In order to design the suitable fractional-order sliding mode controller, a proper switching surface is introduced. Afterward, by using the sliding mode theory and Lyapunov stability theory, a robust fractional-order control law via a nonlinear disturbance observer is proposed to assure the existence of the sliding motion in finite time. The proposed fractional-order sliding mode controller exposes better control performance, ensures fast and robust stability of the closed-loop system, eliminates the disturbances and diminishes the chattering problem. Finally, the effectiveness of the proposed fractional-order controller is depicted via numerical simulation results of practical example and is compared with some other controllers.