Discrete Fractional Order Systems Research Articles

On New Symmetric Fractional Discrete-Time Systems: Chaos, Complexity, and Control

This paper introduces a new symmetric fractional-order discrete system. The dynamics and symmetry of the suggested model are studied under two initial conditions, mainly a comparison of the commensurate order and incommensurate order maps, which highlights their effect on symmetry-breaking bifurcations. In addition, a theoretical analysis examines the stability of the zero equilibrium point. It proves that the map generates typical nonlinear features, including chaos, which is confirmed numerically: phase attractors are plotted in a two-dimensional (2D) and three-dimensional (3D) space, bifurcation diagrams are drawn with variations in the derivative fractional values and in the system parameters, and we calculate the Maximum Lyapunov Exponents (MLEs) associated with the bifurcation diagram. Additionally, we use the C0 algorithm and entropy approach to measure the complexity of the chaotic symmetric fractional map. Finally, nonlinear 3D controllers are revealed to stabilize the symmetric fractional order map’s states in commensurate and incommensurate cases.

Finite-time stability in measure for nabla uncertain discrete linear fractional order systems

Finite-time stability in measure for nabla uncertain discrete linear fractional order systems

Discrete Fractional-Order Systems with Applications in Engineering and Natural Sciences

Discrete Fractional-Order Systems with Applications in Engineering and Natural Sciences

White noise estimation for linear discrete fractional order system

<abstract> <p>The process white noise (PWN) and observation white noise (OWN) estimation problem for linear discrete fractional order systems (LDFOS) is addressed in this study. By using the Grünwald-Letnikov (G-L) operator as a definition of the discrete fractional calculus (DFC), LDFOS is transformed into a class of linear discrete time-delay systems. However, it is different from the general time-delay system, in which the time-delay part is the cumulative sum from time 0 to the previous time. Based on the orthogonal projection theorem, a suboptimal one-step predictor of LDFOS is designed. Due to the existence of cumulative sum time-delay in system, the Riccati equation has one more cumulative sum state error variance term, which is different from the classical Kalman filter (KF). Moreover, using innovation analysis technology, the filtering and fixed-lag smoothing estimators of PWN and OWN in the form of noise orthogonal projection gain matrices are derived. Finally, two simulation examples are given to verify the effectiveness of PWN and OWN estimators.</p> </abstract>

A stabilization approach for a novel chaotic fractional-order discrete neural network

This paper aims to make a contribution to the topic of the stabilization of chaotic dynamics in fractional-order discrete systems by presenting a novel three-dimensional fractional-order discrete neural network formulated by the h-fractional difference operator. A novel theorem is illustrated with the aim of stabilizing the chaotic trajectories of the three-dimensional fractional-order discrete neural network at zero through proposing a quite simple linear control laws. Finally, certain simulation results are carried out to highlight the effectiveness of the stabilization approach proposed herein.

On Dynamics of a Fractional-Order Discrete System with Only One Nonlinear Term and without Fixed Points

Dynamical systems described by fractional-order difference equations have only been recently introduced inthe literature. Referring to chaotic phenomena, the type of the so-called “self-excited attractors” has been so far highlighted among different types of attractors by several recently presented fractional-order discrete systems. Quite the opposite, the type of the so-called “hidden attractors”, which can be characteristically revealed through exploring the same aforementioned systems, is almost unexplored in the literature. In view of those considerations, the present work proposes a novel 3D chaotic discrete system able to generate hidden attractors for some fractional-order values formulated for difference equations. The map, which is characterized by the absence of fixed points, contains only one nonlinear term in its dynamic equations. An appearance of hidden attractors in their chaotic modes is confirmed through performing some computations related to the 0–1 test, largest Lyapunov exponent, approximate entropy, and the bifurcation diagrams. Finally, a new robust control law of one-dimension is conceived for stabilizing the newly established 3D fractional-order discrete system.

Description and analysis of the time–domain response of nabla discrete fractional order systems

AbstractThis paper investigates the time–domain response of nabla discrete fractional order systems by exploring several useful properties of the nabla discrete Laplace transform and the discrete Mittag–Leffler function. In particular, we find the limitation of discrete Mittag–Leffler when adopted in describing the response. Then we establish two fundamental properties of a nabla discrete fractional order system with nonzero initial instant: i) the existence and uniqueness of the system time–domain response; and ii) the dynamic behavior of the zero input response. Finally, a simulation study is provided to show the validity of the theoretical results.

Bifurcation analysis in a discrete time square root response function of predator-prey system with fractional order

Stability and bifurcation analysis for a two spices discrete fractional order system by introducing square root response function of the form is examined in the closed first quadrant . The model parameters h, α, β, μ, σ, η are biologically feasible positive real numbers. Piecewise constant arguments method is applied to obtain the discrete fractional order system, particularly to study the rich dynamical behavior of the proposed system. Because the system has square root response function, the variation matrix of trivial and axial equilibrium states are in-determinant. In order to study the stability of the trivial and semi trivial equilibrium states, the change of variables x(j) = X 2(j) and y(j) = Y 2 (j) is considered. Moreover, we determine the criteria for stability of the interior equilibrium state using jury conditions. The numerical performance is shown for various parameter values and the time series and phase line diagrams are obtained. Bifurcation theory is applied to check whether the system undergoes periodic doubling bifurcations at its semi trivial and interior equilibrium states. Also we explore the periodic halving bifurcation which occurs for the fractional order as a bifurcation parameter. Numerical examples are presented to show the validity and feasibility of the obtained stability criterion in both species, including periodic doubling, periodic - 2, 4, 8, 16, periodic windows and Non periodic orbit (i.e., chaos).

Stability and neimark - sacker bifurcation for a discrete system of one - scroll chaotic attractor with fractional order

This present work investigates the dynamical behaviors of a new form of fractional order three dimensional system with a chaotic attractor of the one-scroll structure and its discretized counterpart. Firstly, existence and parametric conditions for local stability analysis of steady states of the model are addressed. Then the bifurcation theory is applied to investigate the presence of Neimark - Sacker (NS) bifurcations at the coexistence steady state taking the time delay as a bifurcation parameter in the discrete fractional order system. Also the trajectories, phase diagrams, limit cycles, bifurcation diagrams, attractors for period - 1, 2, 3, 4, 5 and a chaotic attractor with one scroll are exhibited for biologically meaningful sets of parameter values in the discretized system. Finally, several numerical examples are presented to assure the validity of the theoretical results and further rich dynamics of the model is explored as well.

Hidden attractors in a new fractional–order discrete system: Chaos, complexity, entropy, and control**The author Adel Ouannas was supported by the Directorate General for Scientific Research and Technological Development of Algeria. The author Shaher Momani was supported by Ajman University in UAE.

This paper studies the dynamics of a new fractional-order discrete system based on the Caputo-like difference operator. This is the first study to explore a three-dimensional fractional-order discrete chaotic system without equilibrium. Through phase portrait, bifurcation diagrams, and largest Lyapunov exponents, it is shown that the proposed fractional-order discrete system exhibits a range of different dynamical behaviors. Also, different tests are used to confirm the existence of chaos, such as 0–1 test and C0 complexity. In addition, the quantification of the level of chaos in the new fractional-order discrete system is measured by the approximate entropy technique. Furthermore, based on the fractional linearization method, a one-dimensional controller to stabilize the new system is proposed. Numerical results are presented to validate the findings of the paper.

Modeling Worm Proliferation in Wireless Sensor Networks with Discrete Fractional Order System

Wireless sensor networks (WSNs) are at risk to cyber attacks and thus security is of vital concern. WSN is a soft target for worm attacks due to fragile defence mechanism in the network . A single unsecured node can essentially propogate the worm in the complete network via communication. Mathematical epidemic models are useful in the study of propagation of worms in WSNs. This work considers a fractional order discrete model of attacking and spreading dynamics of worms in WSNs of the form The proposed epidemic model is probed with the assistance of stability theory. Basic reproduction number (R0)is determined for the analysis of the dynamics of worm propagation in WSNs. The equilibrium states are computed and analyzed the stability. Basic reproduction number R0 enables to discover the threshold values for communication radius and node density distribution. If reproduction number is less than one, the worm free equilibrium state (WFE) is locally asymptotically stable (LAS) and if reproduction number is more than one then the endemic equilibrium state (EE) is asymptotically stable. Numerical illustrations affirm the consistency of the theoretical analysis and stimulating dynamical behavior of the system is observed.

Local and global dynamics of a fractional-order predator–prey system with habitat complexity and the corresponding discretized fractional-order system

This paper is focused on local and global stability of a fractional-order predator–prey model with habitat complexity constructed in the Caputo sense and the corresponding discrete fractional-order system. Mathematical results like positivity and boundedness of the solutions of fractional-order predator–prey model is presented. Conditions for local and global stability of different equilibrium points are proved. It is shown that there may exist fractional-order-dependent instability through Hopf bifurcation. We have determined an extra stability region in the lower range of habitat complexity where all populations coexist in stable state for some fractional-order values but unstable for integer-order value. Dynamics of the discrete fractional-order model is shown to be more complex and depends on both the step-size and fractional-order. It shows Hopf bifurcation, flip bifurcation and chaos with respect to the step-size. Several examples are presented to substantiate the analytical results.

Fractal Control and Synchronization of the Discrete Fractional SIRS Model

SIRS model is one of the most basic models in the dynamic warehouse model of infectious diseases, which describes the temporary immunity after cure. The discrete SIRS models with the Caputo deltas sense and the theories of fractional calculus and fractal theory provide a reasonable and sensible perspective of studying infectious disease phenomenon. After discussing the fixed point of the fractional order system, controllers of Julia sets are designed by utilizing fixed point, which are introduced as a whole and a part in the models. Then, two totally different coupled controllers are introduced to achieve the synchronization of Julia sets of the discrete fractional order systems with different parameters but with the same structure. And new proofs about the synchronization of Julia sets are given. The complexity and irregularity of Julia sets can be seen from the figures, and the correctness of the theoretical analysis is exhibited by the simulation results.

Modeling of discrete fracmemristor and its application

Researches on the fracmemristor have aroused increasing interest in the last several years, but there are no reports on design of the discrete fracmemristor. Based on the fractional-order difference and the mathematical model of the charge-controlled memristor, the discrete fracmemristor is designed where the amount of charge is determined by a fractional-order discrete system. In the numerical simulations, it shows that the pinched hysteresis loops are observed, which imply that the proposed memristor satisfies the definition of the memristor. As an application, the fracmemristor sine map is designed, and multistability is observed regarding the initial conditions of both the memristor and system. It provides a potential model for different applications such as cellular neural networks, modulators, sensors, chaotic systems, and programmable digital circuits.

Study of a discretized fractional-order eco-epidemiological model with prey infection

In this paper, an attempt is made to understand the dynamics of a three-dimensional discrete fractional-order eco-epidemiological model with Holling type II functional response. We first discretize a fractional-order predator-prey-parasite system with piecewise constant arguments and then explore the system dynamics. Analytical conditions for the local stability of different fixed points have been determined using the Jury criterion. Several examples are given to substantiate the analytical results. Our analysis shows that stability of the discrete fractional order system strongly depends on the step-size and the fractional order. More specifically, the critical value of the step-size, where the switching of stability occurs, decreases as the order of the fractional derivative decreases. Simulation results explore that the discrete fractional-order system may also exhibit complex dynamics, like chaos, for higher step-size.

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.

Modelling and simulation of nabla fractional dynamic systems with nonzero initial conditions

AbstractThe paper focuses on the numerical approximation of discrete fractional order systems with the conditions of nonzero initial instant and nonzero initial state. First, the inverse nabla Laplace transform is developed and the equivalent infinite dimensional frequency distributed model of discrete fractional order system is introduced. Then, resorting the nabla discrete Laplace transform, the rationality of the finite dimensional frequency distributed model approaching the infinite one is illuminated. Based on this, an original algorithm to estimate the parameters of the approximate model is proposed with the help of vector fitting method. Additionally, the applicable object is extended from a sum operator to a general system. Three numerical examples are performed to illustrate the applicability and flexibility of the introduced methodology.

On the dynamics, control and synchronization of fractional-order Ikeda map

Abstract This paper is concerned with a fractional Caputo-difference form of Ikeda map. The dynamics of the proposed map is investigated numerically through phase plots and bifurcation diagrams considered from different perspectives. In addition, a stabilization controller is proposed and the asymptotic convergence of the states is established using the stability theory of linear fractional-order discrete systems. Furthermore, a new synchronization scheme is introduced whereby a new 2D fractional-order chaotic map is considered as the master system and the fractional-order Ikeda map is considered as the response system. Experimental investigations and numerical simulations are also provided to confirm the main findings of the study.

Analysis and description of the infinite-dimensional nature for nabla discrete fractional order systems

Fractional order systems in continuous time case are shown to be infinite-dimensional, which is an essential feature used for analysis and synthesis. The objective of this paper is to check the infinite-dimensional characteristic of nabla discrete fractional order systems. Making full use of N-transform, the actual infinite-dimensional state space model is established for both Riemann–Liouville and Caputo cases, which reveals that the finite-dimensional pseudo states are not sufficient to display the transient property of the system and only infinite-dimensional actual states are enough. Apart from these, several interesting points are discussed, including equivalent form, larger order, multiple variable and numerical implementation.

Lyapunov functions for nabla discrete fractional order systems

This paper focuses on the fractional difference of Lyapunov functions related to Riemann–Liouville, Caputo and Grünwald–Letnikov definitions. A new way of building Lyapunov functions is introduced and then five inequalities are derived for each definition. With the help of the developed inequalities, the sufficient conditions can be obtained to guarantee the asymptotic stability of the nabla discrete fractional order nonlinear systems. Finally, three illustrative examples are presented to demonstrate the validity and feasibility of the proposed theoretical results.