Fractional Dynamical Systems Research Articles

Computation of Stability Criterion for Fractional Shimizu–Morioka System Using Optimal Routh–Hurwitz Conditions

Nowadays, the dynamics of non-integer order system or fractional modelling has become a widely studied topic due to the belief that the fractional system has hereditary properties. Hence, as part of understanding the dynamic behaviour, in this paper, we will perform the computation of stability criterion for a fractional Shimizu–Morioka system. Different from the existing stability analysis for a fractional dynamical system in literature, we apply the optimal Routh–Hurwitz conditions for this fractional Shimizu–Morioka system. Furthermore, we introduce the way to calculate the range of adjustable control parameter β to obtain the stability criterion for fractional Shimizu–Morioka system. The result will be verified by using the predictor-corrector scheme to obtain the time series solution for the fractional Shimizu–Morioka system. The findings of this study can provide a better understanding of how adjustable control parameter β influences the stability criterion for fractional Shimizu–Morioka system.

On representation and interpretation of Fractional calculus and fractional order systems

Abstract In this work a relationship between Fractional calculus (FC) and the solution of a first order partial differential equation (FOPDE) is suggested. With this relationship and considering an extra dimension, an alternative representation for fractional derivatives and integrals is proposed. This representation can be applied to fractional derivatives and integrals defined by convolution integrals of the Volterra type, i.e. the Riemann-Liouville and Caputo fractional derivatives and integrals, and the Riesz and Feller potentials, and allows to transform fractional order systems in FOPDE that only contains integer-order derivatives. As a consequence of considering the extra dimension, the geometric interpretation of fractional derivatives and integrals naturally emerges as the area under the curve of a characteristic trajectory and as the direction of a tangent characteristic vector, respectively. Besides this, a new physical interpretation is suggested for the fractional derivatives, integrals and dynamical systems.

Multi-directional and saturated chaotic attractors with many scrolls for fractional dynamical systems

Chaotic dynamical attractors are themselves very captivating in Science and Engineering, but systems with multi-dimensional and saturated chaotic attractors with many scrolls are even more fascinating for their multi-directional features. In this paper, the dynamics of a Caputo three-dimensional saturated system is successfully investigated by means of numerical techniques. The continuity property for the saturated function series involved in the model preludes suitable analytical conditions for existence and stability of the solution to the model. The Haar wavelet numerical method is applied to the saturated system and its convergence is shown thanks to error analysis. Therefore, the performance of numerical approximations clearly reveals that the Caputo model and its general initial conditions display some chaotic features with many directions. Such a chaos shows attractors with many scrolls and many directions. Then, the saturated Caputo system is indeed chaotic in the standard integer case (Caputo derivative order \begin{document}$ \alpha = 1 $\end{document} ) and this chaos remains in the fractional case ( \begin{document}$ \alpha = 0.9 $\end{document} ). Moreover the dynamics of the system change depending on the parameter \begin{document}$ \alpha $\end{document} , leading to an important observation that the saturated system is likely to be regulated or controlled via such a parameter.

Krasnoselskii N-Tupled Fixed Point Theorem with Applications to Fractional Nonlinear Dynamical System

In this paper, N-tupled fixed point theorems for two monotone nondecreasing mappings in complete normed linear space are established. The extension of Krasnoseskii fixed point theorem for a version of N-tupled fixed point is given. Our theoretical results are applied to prove the existence of a mild solution of the system of N-nonlinear fractional evolution equations. Finally, an example of a nonlinear fractional dynamical system is given to illustrate the results.

Controllability of Nonlinear Fractional Langevin Systems

In this paper, we first derive the solution representation of fractional Langevin differential equation represented by the fractional differential coefficient in the sense of Caputo fractional derivative in terms of Mittag-Leffler function. Based on this solution representation, controllability of linear fractional Langevin dynamical systems is studied by using Grammian matrix. Sufficient conditions for the controllability of the nonlinear system are established by using the Schauder’s fixed point theorem. An example is given to verify the results.

A numerical technique for variable-order fractional functional nonlinear dynamic systems

This paper provides an efficient technique to discretize the variable-order fractional functional nonlinear differential equations. The proposed technique is based on a piecewise integro quadratic spline interpolation and finite difference approximation. To reveal the performance and accuracy of the proposed method, the behavioral responses of the suitcase with two wheels and pantograph models with variable-order fractional order are analyzed.

Stability and chaos control of regularized Prabhakar fractional dynamical systems without and with delay

In this paper, we studied the stabilization of nonlinear regularized Prabhakar fractional dynamical systems without and with time delay. We establish a Lyapunov stabiliy theorem for these systems and study the asymptotic stability of these systems without design a positive definite function V (without considering the fractional derivative of function V is negative). We design a linear feedback controller to control and stabilize the nonautonomous and autonomous chaotic regularized Prabhakar fractional dynamical systems without and with time delay. By means of the Lyapunov stability, we obtain the control parameters for these type of systems. We further present a numerical method to solve and analyze regularized Prabhakar fractional systems. Furthermore, by employing numerical simulation, we reveal chaotic attractors and asymptotic stability behaviors for four systems to illustrate the presented theorem.

Necessary conditions of fractional optimal control problems with state constraints in the sense of Riemann‐Liouville

AbstractThis paper is devoted to the optimal control problem of a fractional dynamic system with state constraints in the sense of Riemann‐Liouville. By means of the needle variation, we establish the Pontryagin's maximum principle for the optimal control problem. Moreover, when such a necessary condition is singular in some sense, we investigate the “second‐order” necessary conditions accordingly. As an application, two examples are presented to demonstrate the accuracy and efficiency of the result.

Stability and synchronization criteria for fractional order competitive neural networks with time delays: An asymptotic expansion of Mittag Leffler function

Competitive neural networks(CNNs) has not been well developed in nonlinear fractional order dynamical system, which is developed first time in this paper. Then, by means of a proper Lyapunov functional, asymptotic expansion of Mittag-Leffler function properties, together with some Caputo derivative properties, the testable novel sufficient conditions are given to guarantee the existence, uniqueness of the equilibrium point as well as global asymptotic stability for a class of fractional order competitive neural networks (FOCNNs) are all derived in the form of matrix elements. Furthermore, the boundedness for the solution of FOCNN is presented by employing Cauchy–Schwartz inequality and Gronwall inequality. Besides, a linear feedback control and adaptive feedback control are designed to achieve the global asymptotic synchronization criterion for FOCNNs with time delay and these explored consequences are extended from some previous integer order CNNs output. At last, two numerical simulations are performed to illustrate the effectiveness of our proposed theoretical results.

Perturbation to Noether symmetry for fractional dynamic systems of variable order

Fractional dynamic system of variable order is one of the interesting topics to study. Perturbation to symmetry, which is closely related to its integrability, is worth to be well studied. In this paper, Perturbation to Noether symmetry and adiabatic invariant are investigated in terms of Riemann–Liouville fractional derivative of variable order for fractional generalized Birkhoffian system. Then under some special conditions and transformations, adiabatic invariants for the fractional Birkhoffian system, the fractional Hamiltonian system and the fractional Lagrangian system are discussed. It has been observed that some results obtained from special cases are new and others are consistent with the existing results, which reconfirm the credibility of proposed achievements. Finally, an example is given to illustrate the methods and results.

Adaptive type-2 fuzzy backstepping control of uncertain fractional-order nonlinear systems with unknown dead-zone

A new problem of adaptive type-2 fuzzy fractional control with pseudo-state observer for commensurate fractional order dynamic systems with dead-zone input nonlinearity is considered in presence of unmatched disturbances and model uncertainties; the control scheme is constructed by using the backstepping and adaptive technique. To avoid the complexity of backstepping design process, the dynamic surface control is used. Also, Interval type-2 Fuzzy logic systems (IT2FLS) are used to approximate the unknown nonlinear functions. By using the fractional adaptive backstepping, fractional control laws are constructed; this method is applied to a class of uncertain fractional-order nonlinear systems. In order to better control performance in reducing tracking error, the PSO algorithm is utilized for tuning the controller parameters. Stability of the system is proven by the Mittag–Leffler method. It is shown that the proposed controller guarantees the boundedness property for the system and also the tracking error can converge to a small neighborhood of the origin. The efficiency of the proposed method is illustrated with simulation examples.

Learning Algorithm for Fractional Dynamical Systems with Autocorrelated Errors-in-Variables

In this paper, the stochastic gradient algorithm for learning fractional-order dynamical systems with noisy input and output is proposed. The proposed algorithm allows estimating the parameters of fractional dynamic systems if the input and output noises are color. The proposed algorithm does not require knowledge of the noise distribution laws. The simulation results demonstrate the high accuracy of the proposed learning algorithm in comparison with the least squares learning algorithm.

Application of fractional calculus to combined modified function projective synchronization of different systems

This paper presents the analysis of fractional order dynamical system of combined modified function projective synchronization of different systems. Initially, we formulate the model in fractional order and then investigate their associated properties. We then investigate the chaotic behavior of different systems by considering the fractional order parameter. To obtain the simulation results of the models, we use the Runge-Kutta order four scheme and Adams-Bashforth scheme. The obtained results are discussed in detail for the various values of the fractional order parameters. The obtained graphical results reveal the significance of the fractional order modeling.

Consensus of Multiagent Systems Described by Various Noninteger Derivatives

In this paper, we unify and extend recent developments in Lyapunov stability theory to present techniques to determine the asymptotic stability of six types of fractional dynamical systems. These differ by being modeled with one of the following fractional derivatives: the Caputo derivative, the Caputo distributed order derivative, the variable order derivative, the conformable derivative, the local fractional derivative, or the distributed order conformable derivative (the latter defined in this work). Additionally, we apply these results to study the consensus of a fractional multiagent system, considering all of the aforementioned fractional operators. Our analysis covers multiagent systems with linear and nonlinear dynamics, affected by bounded external disturbances and described by fixed directed graphs. Lastly, examples, which are solved analytically and numerically, are presented to validate our contributions.

Fractal-fractional modelling of partially penetrating wells

In this paper, the fractional order dynamical system theory is used to describe the complex behaviour of partially penetrating wells (PPWs) in a typical reservoir whose geometry is governed by fractal tools. The Green’s function approach, as a generalised impulse response function, is adopted to model the fluid flow in any type of reservoir with a partially penetrating (vertical) well producing from it. Having obtained the initial description of a typical PPW, using the Laplace transform a new dimensionless constant-flow-rate solution is introduced, when wellbore storage and skin effects are significant. The pressure-transient behaviour of a PPW is discussed following two synthetic examples which illustratively depict the effectiveness of the proposed results.

A computational method for solving two‐dimensional nonlinear variable‐order fractional optimal control problems

AbstractIn this paper, a new class of two‐dimensional nonlinear variable‐order fractional optimal control problems (V‐OFOCPs) is introduced where the variable‐order fractional derivative is defined in the Caputo type. The general procedure for solving theses systems is expanding the state variable and the control variable based on the Legendre cardinal functions in the matrix form. Hence, we derive their operational matrix of derivative (OMD) and operational matrix of variable‐order fractional derivative (OMV‐OFD). More significantly, some properties of these basis functions are proved to be exploited in our approach. Using these achieved results, we simply expand the matrix form of the nonlinear performance index in terms of the Legendre cardinal functions and subsequently convert it to an algebraic equation. We emphasize that it is a valuable advantage of applying cardinal functions in approximation theory. Then, we implement the OMD and the OMV‐OFD of the Legendre cardinal functions to transform the variable‐order fractional dynamical system to a system of algebraic equations. Next, the method of constrained extremum is applied to adjoin the constraint equations including the given dynamical system and the initial‐boundary conditions to the performance index by a set of undetermined Lagrange multipliers. Finally, the necessary conditions of the optimality are derived as a system of nonlinear algebraic equations including the unknown coefficients of the state variable, the control variable and the Lagrange multipliers. The applicability and efficiency of the proposed approach are investigated through the various types of test problems.

Controllability of higher order fractional damped delay dynamical systems

The article isconcerned the controllability results for higher order fractional damped delay dynamical systems of any different orders of Caputo derivatives. To establish a necessary and sufficient conditions for the controllability criteria of linear fractional damped delay dynamical systems by Grammian matrix. Sufficient conditions for controllability of the corresponding nonlinear damped delay dynamical systems areobtainedusing successive approximation technique. Examples are includedto verify theoretical results.

Approximation techniques of optimal control problems for fractional dynamic systems in separable Hilbert spaces

We investigate an optimal control problem involving a class of fractional evolution equations in separable Hilbert spaces. The strategy of this paper is establishing low dimensional approximations for this type of equations by using approximation methods. We derive three kinds of convergence results of mild solutions under appropriate assumptions. Then, the convergence result holds for cost functional as well. Further, error estimates of cost functional and optimal controls are obtained. Finally, the proposed concept is supported by an illustrated example.

Fractional Order Version of the Hamilton–Jacobi–Bellman Equation

We consider an extension of the well-known Hamilton–Jacobi–Bellman (HJB) equation for fractional order dynamical systems in which a generalized performance index is considered for the related optimal control problem. Owing to the nonlocality of the fractional order operators, the classical HJB equation, in the usual form, does not hold true for fractional problems. Effectiveness of the proposed technique is illustrated through a numerical example.

Singular points in the solution trajectories of fractional order dynamical systems.

Dynamical systems involving non-local derivative operators are of great importance in Mathematical analysis and applications. This article deals with the dynamics of fractional order systems involving Caputo derivatives. We take a review of the solutions of linear dynamical systems , where the coefficient matrix is in canonical form. We describe exact solutions for all the cases of canonical forms and sketch phase portraits of planar systems. We discuss the behavior of the trajectories when the eigenvalues of matrix are at the boundary of stable region, i.e., . Furthermore, we discuss the existence of singular points in the trajectories of such planar systems in a region of , viz. Region II. It is conjectured that there exists a singular point in the solution trajectories if and only if Region II.