Fractional-order Nonlinear Dynamic Systems Research Articles

Examining reachability of fractional dynamical systems with delays in control utilizing ψ-Hilfer pseudo-fractional derivative

This research focuses on the reachability criteria of fractional dynamical systems with delays in control of order ϑ ∈ (0, 1) and type ϱ ∈ [0, 1] in the context of control theory. The study introduces the utilization of the ψ-Hilfer pseudo fractional derivative to describe the dynamics of systems with delays in control. We obtained the necessary and sufficient conditions for the reachability criteria of fractional order linear dynamical systems by employing the positive definiteness of Gramian matrices. We arrived the sufficient conditions for the reachability criteria of fractional order nonlinear dynamical systems using Banach’s fixed point theorem. A few numerical examples are made available for a better understanding of the theoretical conclusions.

First attempt of barrier functions for Caputo’s fractional-order nonlinear dynamical systems

First attempt of barrier functions for Caputo’s fractional-order nonlinear dynamical systems

Fault-tolerant adaptive fractional controller design for incommensurate fractional-order nonlinear dynamic systems subject to input and output restrictions

In this article, a fault-tolerant adaptive neural network fractional controller has been proposed for a class of uncertain multi-input single-output (MISO) incommensurate fractional-order non-strict nonlinear systems subject to five different types of unknown input nonlinearities, infinite number of actuators failures and arbitrary independent time-varying output constraints. The barrier Lyapunov function (BLF)-based backstepping technique and fractional Lyapunov direct method (FLDM) have been used to design the controller and establish system stability. To tackle the incommensurate derivatives problem, in each step of the backstepping technique, an appropriate Lyapunov function has been selected based on error variables of the considered step and their boundedness has been proved from the last step to the first one. The computational burden is reduced by decreasing the number of adaptive laws. The developed controller satisfies the output constraints and guarantees convergence of the output tracking error to a small neighborhood of the origin and boundedness of all closed-loop signals. Finally, the effectiveness of the designed controller has been demonstrated via simulating three examples.

Fault-Tolerant Adaptive Fractional Controller Design for Incommensurate Fractional-Order Nonlinear Dynamic Systems Subject to Input and Output Restrictions

Fault-Tolerant Adaptive Fractional Controller Design for Incommensurate Fractional-Order Nonlinear Dynamic Systems Subject to Input and Output Restrictions

Dynamic Characteristics Analysis and Chaotic Synchronization of Fractional Order Financial Dynamic System

The equations of time-delay fractional order financial dynamic system are studied. Fractional calculus, as the expansion of the corresponding integer parts, has more complex definition and update laws. It is found that modeling by fractional-order differential equation in some nature phenomenon and engineering field can be consistent with essentiality of the phenomenon and matters, such as in mathematical modeling of viscoelastic materials, electromagnetic wave and other fields. Moreover, there exist more complex dynamical behaviors in fractional-order dynamical system than in integral-order one, and it has more preferable history memory ability. Therefore, it is of theoretical value and practical significance researching the fractional-order nonlinear dynamical systems on the base of the researches on the traditional integral-order nonlinear dynamical system.

Analysis of Fractional-Order Nonlinear Dynamic Systems with General Analytic Kernels: Lyapunov Stability and Inequalities

In this paper, we study the recently proposed fractional-order operators with general analytic kernels. The kernel of these operators is a locally uniformly convergent power series that can be chosen adequately to obtain a family of fractional operators and, in particular, the main existing fractional derivatives. Based on the conditions for the Laplace transform of these operators, in this paper, some new results are obtained—for example, relationships between Riemann–Liouville and Caputo derivatives and inverse operators. Later, employing a representation for the product of two functions, we determine a form of calculating its fractional derivative; this result is essential due to its connection to the fractional derivative of Lyapunov functions. In addition, some other new results are developed, leading to Lyapunov-like theorems and a Lyapunov direct method that serves to prove asymptotic stability in the sense of the operators with general analytic kernels. The FOB-stability concept is introduced, which generalizes the classical Mittag–Leffler stability for a wide class of systems. Some inequalities are established for operators with general analytic kernels, which generalize others in the literature. Finally, some new stability results via convex Lyapunov functions are presented, whose importance lies in avoiding the calculation of fractional derivatives for the stability analysis of dynamical systems. Some illustrative examples are given.

Cluster Synchronization of Fractional-Order Nonlinearly-Coupling Community Networks With Time-Varying Disturbances and Multiple Delays

This paper investigates the adaptive cluster synchronization of fractional-order complex networks with internal and coupling delays as well as time-varying disturbances via the fractional-order hybrid controllers. To be more practical, the unknown disturbances and dynamical behaviors of nodes are assumed to be nonidentical. Based on the properties of fractional calculus and the fractional-order comparison principle, sufficient conditions are derived to guarantee the cluster synchronization of two kinds of fractional-order nonlinear dynamical systems, which extends the results in existing literatures. Numerical simulations are presented to show the effectiveness of our theoretical results.

Integer-fractional decomposition and stability analysis of fractional-order nonlinear dynamic systems using homotopy singular perturbation method

Achieving a simplified model is a major issue in the field of fractional-order nonlinear systems, especially large-scale systems. So that in addition to simplifying the model, the outstanding features of the fractional-order modeling, such as memory feature, are preserved. This paper presented the homotopy singular perturbation method (HSPM) to reduce the complexity of the model and use the advantages of both models of the fractional order and the integer order. This method is a combination of the fractional-order singular perturbation method (FOSPM) and the homotopy perturbation method (HPM). Firstly, the FOSPM is developed for fractional-order nonlinear systems; then, a modification of the HPM is proposed. Finally, the HSPM is presented using a combination of these two methods. fractional-order nonlinear systems can be divided into two lower-order subsystems such as nonlinear or linear integer-order subsystem and linear fractional-order subsystem using this hybrid method. Convergence analysis of tracking error to zero is theoretically presented, and the effectiveness of the proposed method is also evaluated with two examples. Next, the number and location of equilibrium points are compared between the original system and the subsystems obtained from the proposed method. In the end, we show that the stability of fractional-order nonlinear system can be determined by investigating the stability of the subsystems using Theorem 3 and Lemma 2.

Simulation of Chaotic Oscillators of Fractional Order

In 1695 the theory of fractional calculus was introduced, but it only developed as a pure mathematical branch. Currently several research groups have focused on the control, the implementation of filters, PID controllers, synchronization, the implementation of circuits of chaotic systems of fractional order, etc. Currently, the number of applications of fractional calculus is increasing rapidly, these mathematical phenomena have allowed us to describe and model a real object more accurately than the classical "integer" methods. Along with the development of the fractional calculation, it was shown that many fractional-order nonlinear dynamic systems behave in a chaotic manner. This is the type of non-linear systems that are addressed in this research topic with the focus on derivatives of arbitrary order, where numerical simulations of chaotic behavior are presented in non-linear, fractional-order autonomous models. The case studies are six chaotic oscillators of fractional order; The systems of Lorenz, Rӧssler, Financiero, Lui, Chen and Lü, whose attractors are obtained by applying the definitions of the Grünwald-Letnikov definitions and the predictive corrective method of Adams-Bashforth-Moulton.

Homotopy perturbation aided optimization procedure with applications to oscillatory fractional order nonlinear dynamical systems

This paper presents an approximate solution of nonlinear fractional differential equations (FDEs) that exhibit an oscillatory behavior by using a metaheuristic technique. The solutions of the governing equations are approximated by using homotopy perturbation method (HPM) along with the fractional derivative in the Caputo sense. The designed methodology is based on a weighted series of HPM in conjunction with a nature-inspired algorithm. The idea is instantly fascinated by the researchers on the consequent implementation of nature-inspired learning algorithms such as a Cuckoo search algorithm (CSA). The usage of CSA has accelerated the minimized search path of error to the convergent values of the solution. The validity and accuracy of the proposed technique are ascertained by calculating the approximate solution and the error norms which ensure the convergence of the approximation that can be further increased. The critical analysis is also provided by the numerical simulation of two different test models. Discussion of key points has been determined by the tabulation of numerical values and graphs. Comparative study of the results with known numerical technique is also performed.

Generalized Wright stability for distributed fractional-order nonlinear dynamical systems and their synchronization

In this article, we present a generalization of stability theorems for Caputo fractional derivative to the distributed fractional-order (DFO) case by using the Laplace transform and the asymptotical expansion of the generalized Mittag–Leffler function. We propose the definition of the generalized Wright stability to study the stability of DFO nonlinear dynamical system using Lyapunov direct method. The linear feedback control is used to stabilize a class of chaotic DFO nonlinear dynamical systems. Using Lyapunov direct method, we study the synchronization between two identical chaotic systems and between two other different in the linear terms. The chaotic DFO Lorenz system is given as an example to achieve the linear feedback control technique. Another two examples which are chaotic DFO complex Chen and Lu systems are used to show the validity and feasibility of our proposed synchronization scheme. Numerical simulations are implemented to verify the results of these investigations.

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.

Secure communication and image encryption scheme based on synchronisation of fractional order chaotic systems using backstepping

Synchronisation strategy of fractional order nonlinear dynamical systems is utilised to address the problem of secure communication by introducing additional security features. Initially, a backstepping-based controller is designed for synchronisation of two similar fractional order chaotic systems in master-slave configuration. The controller obtained is divided into two parts with one part transmitted through the communication channel whereas the other part has to be designed at the receiver end so as to obtain synchronisation between master and slave system. Further, only two signals are transmitted out of which one has an embedded message signal and the other is a complex combination of master states which will be used for synchronisation purpose at the receiver end for recovery of message signal. The initial conditions, system parameters, fractional orders, the structure of controller and the combination of master system states form a large key set which should be known is advance in order to breach the security. Here, the scheme for secure transmission of speech signal is presented first and further a scheme for image encryption is given. The simulation results and security analysis using different quantitative measures are presented at the end to demonstrate the applicability and effectiveness of the proposed scheme.

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.

New Results on Exponential Stability of Fractional Order Nonlinear Dynamic Systems

In this letter stability analysis of fractional order nonlinear systems is studied. An extension of Lyapunov direct method for fractional order systems is proposed by using the properties of Mittag-Leffler function and Laplace transform. Some new sufficient conditions which ensure local exponential stability of fractional order nonlinear systems are proposed firstly. And we apply these conditions to the Riemann-Liouville fractional order systems by using fractional comparison principle. Finally, three examples are provided to illustrate the validity of the proposed approach.

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.

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 Lu 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.

Generalized Mittag-Leffler stability of multi-variables fractional order nonlinear systems

In this paper, the stability of multi-variable fractional order nonlinear dynamic system is investigated. We propose the definition of generalized Mittag-Leffler stability with multi-variable and introduce the fractional Lyapunov direct method with multi-variable. Meanwhile, a novel approach is suggested to study generalized Mittag-Leffler stability in multi-variable fractional order nonlinear dynamic systems. An interesting multi-variable fractional order Lotka–Volterra predator–prey model is used to illustrate the proposed method and its effectiveness.

Robust synchronization for a class of fractional-order chaotic and hyperchaotic systems

In this paper, the issue of robust synchronization for a class of fractional-order chaotic and hyperchaotic systems with model uncertainties and disturbances is studied. A stability criterion for fractional-order nonlinear dynamic systems is introduced, and an adaptive scheme is contrived to accomplish synchronization of fractional-order chaotic and hyperchaotic systems. The controller contains only a single state variable, which is simple and flexible in implementation. Two corresponding numerical examples are given to confirm the theoretical results of the paper.

Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability

Stability of fractional-order nonlinear dynamic systems is studied using Lyapunov direct method with the introductions of Mittag–Leffler stability and generalized Mittag–Leffler stability notions. With the definitions of Mittag–Leffler stability and generalized Mittag–Leffler stability proposed, the decaying speed of the Lyapunov function can be more generally characterized which include the exponential stability and power-law stability as special cases. Finally, four worked out examples are provided to illustrate the concepts.