Discrete Fractional Order Research Articles

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.

Oscillation Analysis for Nonlinear Discrete Fractional Order Delay and Neutral Equations with Forcing Term

Oscillation Analysis for Nonlinear Discrete Fractional Order Delay and Neutral Equations with Forcing Term

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.

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.

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.

Leader-Following Consensus Control of Nabla Discrete Fractional Order Multi-Agent Systems

This paper studies a consensus problem for discrete-time linear nabla fractional order multi-agent systems with Riemann-Liouville difference operator. With the help of the discrete fractional Lyapunov direct method, a state feedback stabilization problem of a discrete-time linear nabla fractional order system is firstly analyzed. Then a distributed consensus control law is proposed for a discrete-time linear nabla fractional order multi-agent system. Some sufficient conditions are presented to guarantee that the leader-following consensus can be achieved by the proposed algorithm. The control gain is determined according to an algebraic Riccati inequality. Finally, simulation results are presented to demonstrate the effectiveness of theoretical analysis.

Oscillation Theorems for Certain Forced Nonlinear Discrete Fractional Order Equations

The main objective of this work is to obtain some new sufficient conditions that are essential for the oscillation of the solutions of forced nonlinear discrete fractional equations of the form \begin{align*} \Delta\left[\Delta^\mu(u(j))\right]+\eta(j)\Phi(u(j))=\psi(j), j\in N_0 \end{align*} where \(\Delta^{\mu-1}u(0)=u_0\); \(\Delta u(j)=u(j+1)-u(j)\) and \(\Delta^\mu\) is defined as the difference operator of the Riemann-Liouville (R-L) derivative of order \(\mu\in(0,1]\) and \(N_0=\{0,1,2,\cdots\}\). Numerical examples are presented to show the validity of the theoretical results.

Dynamical analysis of a discrete conformable fractional order bacteria population model in a microcosm

In this paper, conformable fractional order differential equations with piecewise constant arguments are used for a modeling population density of a bacteria species in a microcosm. The discretization process of a differential equation with piecewise constant arguments gives us two dimensional discrete dynamical system in the interval t∈[nh,(n+1)h). By using the center manifold theorem and the bifurcation theory, it is shown that the discrete dynamical system undergoes flip and Neimark–Sacker bifurcation depending on the parameter r. It is observed that the model describes some biological phenomena for a bacteria population such as homogeneous bacteria distributions and inhomogeneous spatial population distributions. In addition, the effect of fractional order derivative on dynamic behavior of the system is investigated. Finally, all theoretical results are supported by numerical simulations.

Bifurcation and dynamical behaviour of a fractional order Lorenz-Chen-Lu like chaotic system with discretization

In this article, the dynamical behaviour of a discrete fractional order Lorenz-Chen-Lu like chaotic system is investigated. A discretization process is used to obtain a discrete version. The system has three constants, three fixed points and generates typical chaotic attractors. Numerical verifications are presented for the analysis of dynamical system. The chaotic attractors, bifurcation diagrams and phase portraits are provided for distinctive values of parameters. The sensitivity to initial conditions of the parameter values for the fixed points is also discussed.

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.

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.

Discrete fractional order SIR epidemic model and it’s stability

This paper deals with the fractional order SIR epidemic model in discrete form describing the dynamics of disease propagation in a population. The Disease Free Equilibrium (DFE) and Endemic Equilibrium (EE) points are computed and the stability nature at these points are discussed. Applying next generation matrix method to calculated the basic reproduction number R0. The time plots, phase portraits and bifurcation are presented for different sets of parameter values. The numerical simulations of the discrete fractional-order SIR epidemic model are performed to illustrate the stability of the epidemic model.

Allee effect and Holling type - II response in a discrete fractional order prey - predator model

A gauss type prey - predator model is considered with Allee effect and Holling type II response. Fractional order two species system is discretized. The discretized system exhibits much richer and complex dynamics than its corresponding continuous version.Bifurcation types like flip, neimark sacker and chaos exist in the discretized system.Existence of the positive fixed points is established and local stability of discrete fractional order system is discussed with the variational matrix. It is also shown that the system allows a flip bifurcation and a neimark - sacker bifurcation. Rich dynamical nature of the system is established through time plots with phase trajectory, exciting invariant circles and periodic doubling bifurcation which leads to chaos.

Behavior of a Discrete Fractional Order SIR Epidemic Model

In this paper we investigate the dynamical behavior of a SIR epidemic model of fractional order. Disease Free Equilibrium point, Endemic Equilibrium point and basic reproductive number are obtained. Time series plots, phase portraits and bifurcation diagrams are presented for suitable parameter values. Also some numerical examples are provided to illustrate the dynamics of the system.

Numerical Analysis of a Fractional Order Discrete Prey – Predator System with Functional Response

This study presents numerical examples of Discrete Fractional Order Prey Predator interactions with Functional Response. The process of discretization is applied and the version of discrete equations is obtained. Fixed points are determined and the stability around the fixed points is analyzed. Also the theoretical analysis has been verified from the numerical simulations, which help better understanding of the proposed system. Rich dynamics of system is exhibited by Bifurcation diagram and Periodic Oscillations for suitable parameters values.

Bifurcation and mixed tracking of the discrete fractional LPA model

The memory effect in fractional calculus has made it a more realistic approach for the practical modeling of real life phenomena ranging from population, electric circuits, etc. In this paper, a discrete fractional order model for the Larva–Pupa–Adult (LPA) beetle population dynamics is proposed. The bifurcation of the proposed model showed novel dynamics. The mixed tracking control of the discrete fractional order LPA model was also considered to adjust individually and independently the population of each of larva, pupa or adult.

Fractional order unknown input filter design for fault detection of discrete fractional order linear systems

In this study, a new method is introduced to design an estimator for discrete-time linear fractional order systems, which are affected by unknown disturbances. The main goal of this study is decoupling disturbance and uncertainties from the true states for discrete fractional order systems in noisy environment. The fractional Kalman filter framework is exploited to develop a robust estimator against unknown inputs (UIs) in noisy environment. The proposed filter is exploited to detect faults in fractional order systems. Simulation results illustrate the advantages of this robust filter for state estimation and fault detection of fractional order model of ultra-capacitor (UC). The robustness of the designed filter is shown in the sense of disturbance decoupling in the presence of noise.

Discrete state space systems of fractional order

The present article devotes to the study of linear time invariant discrete fractional order state space systems using a novel approach. First, we replace the conventional Grünwald-Letnikov-type backward difference operator with the equivalent Riemann-Liouville-type nabla difference operator and obtain the system response using variation of constants and discrete Laplace transform methods. Next, we present three different algorithms to construct state transition matrix of the system. Finally, we provide an example to illustrate the applicability of established result.

Discrete state space systems of fractional order

The present article devotes to the study of linear time invariant discrete fractional order state space systems using a novel approach. First, we replace the conventional Grunwald-Letnikov-type backward difference operator with the equivalent Riemann-Liouville-type nabla difference operator and obtain the system response using variation of constants and discrete Laplace transform methods. Next, we present three different algorithms to construct state transition matrix of the system. Finally, we provide an example to illustrate the applicability of established result.