Fractional-order Dynamical Systems Research Articles

The Framework of Mechanics for Dynamic Behaviors of Fractional-Order Dynamic Systems

The Framework of Mechanics for Dynamic Behaviors of Fractional-Order Dynamic Systems

Bifurcation analysis of fractional-order VD model

In this paper, we have introduced the fractional-order VD model. Also, we studied the basic results such as existence, uniqueness, boundedness of the fractional-order dynamical system. The local stability and Hopf bifurcation analysis of the fractional-order VD model are studied. Finally, we present some numerical simulations that validates the theoretical results.

Fractional Order System Identification With Occupation Kernel Regression

While fractional order systems have been employed broadly throughout science and engineering, system identification aimed at nonlinear fractional order dynamical systems remain in their infancy. One reason for this is that local estimates cannot be used to obtain a sample of fractional order dynamics in the same way that is done for integer order systems. This letter leverages occupation kernels to poise a trajectory as the fundamental unit of data from a fractional order dynamical system. When combined with a regularized regression problem, an approximation of fractional order dynamics is obtained as a linear combination of occupation kernels. A battery of numerical experiments are executed to validate the developed method, and it is demonstrated over two dynamical systems that accurate estimates of fractional order dynamics can be obtained both along trajectories and also nearby regions.

Nonexistence of invariant manifolds in fractional-order dynamical systems

Invariant manifolds are important sets arising in the stability theory of dynamical systems. In this article, we take a brief review of invariant sets. We provide some results regarding the existence of invariant lines and parabolas in planar polynomial systems. We provide the conditions for the invariance of linear subspaces in fractional order systems. Further, we provide an important result showing the nonexistence of invariant manifolds (other than linear subspaces) in fractional order systems.

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.

A non-integer sliding mode controller to stabilize fractional-order nonlinear systems

In this study, we examine the stabilization of fractional-order chaotic nonlinear dynamical systems with model uncertainties and external disturbances. We used the sliding mode controller by a new approach for controlling and stabilization of these systems. In this research, we replaced a continuous function with the sign function in the controller design and the sliding surface to suppress chattering and undesirable vibration effects. The advantages of the proposed control method are rapid convergence to the equilibrium point, the absence of chattering and unwanted oscillations, high resistance to uncertainties, and the possibility of applying this method to most fractional order chaotic systems. We applied the direct method of Lyapunov stability theory and the frequency distributed model to prove the stability of the slip surface and closed loop system. Finally, we simulated this method on two commonly used and practical chaotic systems and presented the results.

Dynamic characteristics analysis of time-delay fractional order dynamic system

In this paper, a kind of time-delay fractional order dynamic system is studied. The classical integral sliding mode control method is used to design a single controller to control the time-delay fractional order Liu system from chaos to fixed point, and the single controller is further designed to control the time-delay fractional order Liu system from chaotic attractor to limit loop, so the chaos control of time-delay fractional order nonlinear dynamic system is achieved. On that basis, the dynamic characteristics of the time-varying delayed fractional-order Lorenz system are analyzed by the simulation, and the chaotic phenomena of the time-varying delayed fractional-order Lorenz system are verified under the provided time delay and order.

Optimal Homotopy Asymptotic Method-Least Square for Solving Nonlinear Fractional-Order Gradient-Based Dynamic System from an Optimization Problem

In this paper, we consider an approximate analytical method of optimal homotopy asymptotic method-least square (OHAM-LS) to obtain a solution of nonlinear fractional-order gradient-based dynamic system (FOGBDS) generated from nonlinear programming (NLP) optimization problems. The problem is formulated in a class of nonlinear fractional differential equations, (FDEs) and the solutions of the equations, modelled with a conformable fractional derivative (CFD) of the steepest descent approach, are considered to find the minimizing point of the problem. The formulation extends the integer solution of optimization problems to an arbitrary-order solution. We exhibit that OHAM-LS enables us to determine the convergence domain of the series solution obtained by initiating convergence-control parameter Cj′s. Three illustrative examples were included to show the effectiveness and importance of the proposed techniques.

Stability analysis of the fractional-order prey-predator model with infection

ABSTRACT In this paper, we propose a fractional-order prey-predator model with infection on both populations. First, we prove some important results such as existence, uniqueness, non-negativity and boundedness of the solutions of the fractional-order dynamical system. Next, we discuss the local stability and global stability of the fractional-order prey-predator model. Numerical simulations are presented with several examples.

Mixed-Mode Oscillations Based on Complex Canard Explosion in a Fractional-Order Fitzhugh-Nagumo Model.

Abstract This article highlights particular mixed-mode oscillations (MMO) based on canard explosion observed in a fractional-order Fitzhugh-Nagumo (FFHN) model. In order to rigorously analyze the dynamics of the FFHN model, a recently introduced mathematical notion, the Hopf-like bifurcation (HLB), which provides a precise definition for the change between a fixed point and an S−asymptotically T−periodic solution, is used. The existence of HLB in this FFHN model is proved and the appearance of MMO based on canard explosion in the neighborhoods of such HLB points are numerically investigated using a new algorithm: the global-local canard explosion search algorithm. This MMO is constituted of various patterns of solutions with an increasing number of small-amplitude oscillations when two key parameters of the FFHN model are varied simultaneously. On the basis of such numerical experiment, it is conjectured that chaos could occur in a two-dimensional fractional-order autonomous dynamical system, with the fractional-order close to one. Therefore, this very simple two-dimensional FFHN model, presents an incredible ability to mimic the complex dynamics of neurons.

Mittag–Leffler stability of nabla discrete fractional-order dynamic systems

In the present study, the definition of discrete Mittag–Leffler stability is derived to characterize convergence rule of the pseudostates for nabla discrete fractional-order dynamic systems. Applying the Lyapunov stability theory, some new criteria are proposed to determine asymptotic stability of the zero equilibrium. In addition, by applying fractional comparison principle, the results are extended from Caputo discrete fractional-order systems to Riemann–Liouville systems. Moreover, a useful inequality is proposed to further improve the availability of the presented methods. Finally, some meticulously designed simulations are provided to verify the correctness and practicability of the elaborated stability notions.

CRONE Control Strategy for Air Pressure System

Pressure process in industry is inevitable and is considered with foremost importance in many processes like boilers, pressure vessels, reactors, closed vessels, steam drums and flash points. Pressure measurement and control is important requirements of all process, since it is related to safety of the human resources and equipment. So in this work an Air Pressure System (APS) is considered for investigation. Over the years an increasing attention made on fractional order controller and dynamic systems using fractional order calculus. Its theoretical and practical interests have been confirmed today, and its relevance to engineering society can be think carefully about as developing scientific avenue. This project discusses the design and implementation of generations of CRONE controller for APS. The Mathematical model of APS is determined from the various operating regions by using worst case modeling technique. The three generations of the strategy of CRONE control are designed based on worst case model. Simulation runs of set point tracking responses are documented. The time domain and integral error indices performances are obtained and compared on strategies of the generation CRONE controller. It is noticed that Third Generation CRONE (TGC) controller outperforms provide better performance than the other generations of CRONE controller.

Lyapunov exponents spectrum estimation of fractional order nonlinear systems using Cloned Dynamics

This work presents the determination of Lyapunov spectrum of fractional order dynamical systems. The method considers the Cloned Dynamic technique, which is based on time evaluation of exact copies of dynamic equations with small initial conditions disturbances. As its calculations are simple, this method can be applied to various dynamic systems types, in particular fractional order, nonlinear and time varying systems, and non linear systems where Jacobian are impossible to evaluate. The predictor-corrector Adams-Bashforth-Moulton algorithm is used for fractional order system numerical evaluation. All Lyapunov exponents are estimated based on the convergence or divergence of the small disturbed clones with respect to the reference (fiducial) trajectory and application of Gram-Schmidt Reorthonormalization after each convenient periodic time interval. To evaluate the application of this method, three third order dynamical systems are explored: Jerk, Financial, and a Four Wing systems. Integer order, commensurate fractional order and incommensurate fractional order are explored. To confirm chaotic or non-chaotic behavior, 0-1 Test is used and Jerk system is studied with a classical method for Lyapunov exponents determination also. The advantages and contribution of the proposed method the possibility to study incommensurate orders fractional cases and explore non linear systems without need of Jacobian determination, as is proved with applications shown. The Lyapunov Exponents and the phase portraits were simulated together to validate this method and the obtained results are in good agreement.

Some properties of Sadik transform and its applications of fractional-order dynamical systems in control theory

In this paper, we study some new properties of Sadik transform such as integration, time delay, initial value theorem, and final value theorem. Moreover, we prove the theorem of Sadik transform for Caputo fractional derivative and we also establish sufficient conditions for the existence of the Sadik transform of Caputo fractional derivatives. At the end, the fractional-order dynamical systems in control theory as an application of this transform is discussed, in addition, some numerical examples to justify our results.

Chaos control and solutions of fractional-order Malkus waterwheel model

Abstract Malkus waterwheel model is a Lorenz type chaotic-physical model expressed in terms of a system of nonlinear ordinary differential equations. In this investigation, we consider fractional-order Malkus waterwheel model via Caputo type time derivative and present chaos control, anti-synchronization, numerical solutions of the fractional system. We also associate fractional-order Malkus model with two different optimal control problems. Computational results indicate that this study may serve as a framework for chaotic behavior analysis and approximate solutions of many different parametric systems. The paper may be considered as a novel contribution because optimal control formulations, numerical solutions, stability analysis for fractional-order Malkus model are studied first time in this paper. This research work may be useful for researchers concerning with chaos analysis and approximate solutions of fractional-order chaotic dynamical systems.

High-performance modeling and discrete-time sliding mode control of uncertain non-commensurate linear time invariant MIMO fractional order dynamic systems

This paper primarily proposes an analytical solution to the set of coupled non-commensurate linear time invariant fractional order differential equations (representing Multi input Multi output (MIMO) fractional order dynamic systems). The proposed solution is reviewed and accurately truncated to have a short finite-memory and to demonstrate further compatibility for practical applications. It is also stated that this discrete-time approach is explicitly superior to any approach in continuous-time in terms of the possibility of implementation. Standards are provided for adequate truncation of the response with respect to the application. A novel discrete-time enhanced sliding mode control (ESMC) scheme is proposed for MIMO fractional order dynamic systems that can cope with parameter uncertainty while illustrating robustness against disturbance. Due to its structure, ESMC shows no sign of chattering in its control signal and employs no fractional operator in its structure; hence, computational cost of ESMC is minuscule. A novel mathematical method is also proposed for acquiring the stability bounds and solving linear matrix inequalities that arise in this problem. A famous MIMO fractional order dynamic system is presented and controlled under considerable disturbance and parameter uncertainty to render the merits of ESMC.

Controlling Chaos for a Fractional-Order Discrete System

In this article two different controllers for the stabilization of a fractional-order discrete system in the left Caputo discrete delta operator sense are given. The first one acts by a fractional proportional pulse control, the second acts by a fractional feedback control. These controllers are applied to fractional-order chaotic discrete dynamical systems to obtain their stability and also we show a comparison with the integer order dynamical systems stability. Some simulations are presented for the fractional logistic map and the fractional Henon map.

Tracking Control for Fractional Order Systems with High Relative Degree Outputs

The fractional order systems represented in commensurate state-space form can exhibit loss of controllability in the error dynamics due to variation in the order of derivative. This poses a challenge while designing tracking control. In this paper we investigate this phenomena, define the concept of relative degree for fractional order systems and also develop new set of identities which simplify the analysis and design of controllers for fractional order dynamical systems.

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.

Parametric Identification of Nonlinear Fractional Hammerstein Models

In this paper, a system identification method for continuous fractional-order Hammerstein models is proposed. A block structured nonlinear system constituting a static nonlinear block followed by a fractional-order linear dynamic system is considered. The fractional differential operator is represented through the generalized operational matrix of block pulse functions to reduce computational complexity. A special test signal is developed to isolate the identification of the nonlinear static function from that of the fractional-order linear dynamic system. The merit of the proposed technique is indicated by concurrent identification of the fractional order with linear system coefficients, algebraic representation of the immeasurable nonlinear static function output, and permitting use of non-iterative procedures for identification of the nonlinearity. The efficacy of the proposed method is exhibited through simulation at various signal-to-noise ratios.