scholarly journals Design, Analysis and Comparison of a Nonstandard Computational Method for the Solution of a General Stochastic Fractional Epidemic Model

Axioms ◽  
2021 ◽  
Vol 11 (1) ◽  
pp. 10
Author(s):  
Nauman Ahmed ◽  
Jorge E. Macías-Díaz ◽  
Ali Raza ◽  
Dumitru Baleanu ◽  
Muhammad Rafiq ◽  
...  
Keyword(s):  
Finite Difference ◽  
Fractional Order ◽  
Epidemic Model ◽  
Equilibrium Points ◽  
Fractional Order System ◽  
Computational Method ◽  
Order System ◽  
Parametric Perturbation ◽  
Nonstandard Finite Difference ◽  
General Stochastic

Malaria is a deadly human disease that is still a major cause of casualties worldwide. In this work, we consider the fractional-order system of malaria pestilence. Further, the essential traits of the model are investigated carefully. To this end, the stability of the model at equilibrium points is investigated by applying the Jacobian matrix technique. The contribution of the basic reproduction number, R0, in the infection dynamics and stability analysis is elucidated. The results indicate that the given system is locally asymptotically stable at the disease-free steady-state solution when R0<1. A similar result is obtained for the endemic equilibrium when R0>1. The underlying system shows global stability at both steady states. The fractional-order system is converted into a stochastic model. For a more realistic study of the disease dynamics, the non-parametric perturbation version of the stochastic epidemic model is developed and studied numerically. The general stochastic fractional Euler method, Runge–Kutta method, and a proposed numerical method are applied to solve the model. The standard techniques fail to preserve the positivity property of the continuous system. Meanwhile, the proposed stochastic fractional nonstandard finite-difference method preserves the positivity. For the boundedness of the nonstandard finite-difference scheme, a result is established. All the analytical results are verified by numerical simulations. A comparison of the numerical techniques is carried out graphically. The conclusions of the study are discussed as a closing note.

scholarly journals A New Fractional-Order Chaotic System with Different Families of Hidden and Self-Excited Attractors

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 564 ◽  
Author(s):  
Jesus Munoz-Pacheco ◽  
Ernesto Zambrano-Serrano ◽  
Christos Volos ◽  
Sajad Jafari ◽  
Jacques Kengne ◽  
...  
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Chaotic Attractor ◽  
Equilibrium Points ◽  
Fractional Order System ◽  
Order System ◽  
Chaotic Attractors ◽  
Equilibrium System ◽  
Hidden Attractors ◽  
Hidden Chaotic Attractor

In this work, a new fractional-order chaotic system with a single parameter and four nonlinearities is introduced. One striking feature is that by varying the system parameter, the fractional-order system generates several complex dynamics: self-excited attractors, hidden attractors, and the coexistence of hidden attractors. In the family of self-excited chaotic attractors, the system has four spiral-saddle-type equilibrium points, or two nonhyperbolic equilibria. Besides, for a certain value of the parameter, a fractional-order no-equilibrium system is obtained. This no-equilibrium system presents a hidden chaotic attractor with a `hurricane’-like shape in the phase space. Multistability is also observed, since a hidden chaotic attractor coexists with a periodic one. The chaos generation in the new fractional-order system is demonstrated by the Lyapunov exponents method and equilibrium stability. Moreover, the complexity of the self-excited and hidden chaotic attractors is analyzed by computing their spectral entropy and Brownian-like motions. Finally, a pseudo-random number generator is designed using the hidden dynamics.

scholarly journals Dynamics Analysis and Control of a Five-Term Fractional-Order System

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Li-xin Yang ◽  
Xiao-jun Liu
Keyword(s):  
Fractional Order ◽  
Equilibrium Points ◽  
Fractional Order System ◽  
Phase Portraits ◽  
Order System ◽  
Bifurcation Diagrams ◽  
Compound Structure ◽  
Dynamical Properties ◽  
The Stability ◽  
And Control

This paper proposes a new fractional-order chaotic system with five terms. Firstly, basic dynamical properties of the fractional-order system are investigated in terms of the stability of equilibrium points, Jacobian matrices theoretically. Furthermore, rich dynamics with interesting characteristics are demonstrated by phase portraits, bifurcation diagrams numerically. Besides, the control problem of the new fractional-order system is discussed via numerical simulations. Our results demonstrate that the new fractional-order system has compound structure.

Dynamical Analysis of a Fractional-Order Hantavirus Infection Model

2020 ◽  
Vol 21 (2) ◽  
pp. 171-181 ◽  
Author(s):  
Mahmoud Moustafa ◽  
Mohd Hafiz Mohd ◽  
Ahmad Izani Ismail ◽  
Farah Aini Abdullah
Keyword(s):  
Fractional Order ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Fractional Order System ◽  
Order System ◽  
Logistic Growth ◽  
Hantavirus Infection ◽  
Infection Model ◽  
The Impact

AbstractThis paper considers a Hantavirus infection model consisting of a system of fractional-order ordinary differential equations with logistic growth. The fractional-order model describes the spread of Hantavirus infection in a system consisting of a population of susceptible and infected mice. The existence, uniqueness, non-negativity and boundedness of the solutions are established. In addition, the local and global asymptotic stability of the equilibrium points of the fractional order system and the basic reproduction number are studied. The impact of basic reproduction number and carrying capacity on the stability of the fractional order system are also theoretically and numerically investigated.

scholarly journals Bifurcations of a New Fractional-Order System with a One-Scroll Chaotic Attractor

2019 ◽  
Vol 2019 ◽  
pp. 1-15
Author(s):  
Xiaojun Liu ◽  
Ling Hong ◽  
Lixin Yang ◽  
Dafeng Tang
Keyword(s):  
Fractional Order ◽  
Chaotic Attractor ◽  
System Parameter ◽  
Equilibrium Points ◽  
Fractional Order System ◽  
Order System ◽  
Order Case ◽  
The Stability ◽  
The One ◽  
Derivative Order

In this paper, a new fractional-order system which has a chaotic attractor of the one-scroll structure is presented. Firstly, the stability of the equilibrium points of the system is investigated. And based on the stability analysis, the generation conditions of the one-scroll structure for the attractor are determined. In a commensurate-order case, bifurcations with the variation of a system parameter are investigated as derivative orders decrease from 0.99. In an incommensurate-order case, bifurcations with the variation of a derivative order are analyzed as other orders decrease from 1.

scholarly journals Dynamics of a SIR Epidemic Model of Childhood Diseases with a Saturated Incidence Rate: Continuous Model and Its Nonstandard Finite Difference Discretization

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1459
Author(s):  
Isnani Darti ◽  
Agus Suryanto
Keyword(s):  
Finite Difference ◽  
Epidemic Model ◽  
Equilibrium Points ◽  
Continuous Model ◽  
Sir Epidemic Model ◽  
Childhood Diseases ◽  
Sir Epidemic ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
Nonstandard Finite Difference

A SIR epidemic model that describes the dynamics of childhood disease with a saturated incidence rate and vaccination program at a constant rate was investigated. For the continuous model we first show its basic properties, namely, the non-negativity and boundedness of solutions. Then we investigate the existence and both local and global stability of the equilibrium points. It was found that the existence and stability properties of equilibrium points fully determined the basic reproduction number. We also propose and analyze a discrete-time analogue of the continuous childhood diseases by applying a nonstandard finite difference method. It is shown that our discrete model preserves the dynamical properties of the corresponding continuous model, such as the positivity solutions, the population conservation law, the existence of equilibrium points and their global stability properties.

Stabilization of Unstable Fixed Points of Fractional-Order Systems by Fractional-Order Linear Controllers and Its Applications in Suppression of Chaotic Oscillations

2010 ◽  
Vol 132 (2) ◽  
Author(s):  
Mohammad Saleh Tavazoei ◽  
Mohammad Haeri
Keyword(s):  
Fractional Order ◽  
Equilibrium Points ◽  
Output Feedback Control ◽  
Fractional Order System ◽  
Dynamic State ◽  
Order System ◽  
Chen System ◽  
Order Linear ◽  
Linearized Model ◽  
Linear Controllers

In this paper, two fractional-order linear controllers are proposed to stabilize unstable equilibrium points of a chaotic fractional-order system. The first controller is based on the dynamic output feedback control idea and requires detectability of the linearized model of the fractional-order system on the equilibrium point. The second controller is a dynamic state feedback controller and requires observability of the linearized model. In both considered cases, the stabilizability of the model is assumed. The number of inner states in the second controller is one and therefore its structure is much simpler than the first controller. To illustrate the applicability, these controllers are applied to control chaos in the fractional-order Chen system. Numerical simulations results are presented to evaluate the performance of the proposed controllers.

scholarly journals The Dynamics and Synchronization of a Fractional-Order System with Complex Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Xiaoya Yang ◽  
Xiaojun Liu ◽  
Honggang Dang ◽  
Wansheng He
Keyword(s):  
Fractional Order ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Complex Variables ◽  
Fractional Order System ◽  
Order System ◽  
Chaotic Attractors ◽  
Fractional Order Systems ◽  
System Parameters ◽  
The Stability

A fractional-order system with complex variables is proposed. Firstly, the dynamics of the system including symmetry, equilibrium points, chaotic attractors, and bifurcations with variation of system parameters and derivative order are studied. The routes leading to chaos including the period-doubling and tangent bifurcations are obtained. Then, based on the stability theory of fractional-order systems, the scheme of synchronization for the fractional-order complex system is presented. By designing appropriate controllers, the synchronization for the system is realized. Numerical simulations are carried out to demonstrate the effectiveness of the proposed scheme.

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