Complex dynamics of a fractional-order SIR system in the context of COVID-19

It is found that the fractional order memristor model can better simulate the characteristics of memristors and that chaotic circuits based on fractional order memristors also exhibit abundant dynamic behavior. This paper proposes an active fractional order memristor model and analyzes the electrical characteristics of the memristor via Power-Off Plot and Dynamic Road Map. We find that the fractional order memristor has continually stable states and is therefore nonvolatile. We also show that the memristor can be switched from one stable state to another under the excitation of appropriate voltage pulse. The volt–ampere hysteretic curves, frequency characteristics, and active characteristics of integral order and fractional order memristors are compared and analyzed. Based on the fractional order memristor and fractional order capacitor and inductor, we construct a chaotic circuit, of which the dynamic characteristics with respect to memristor’s parameters, fractional order α, and initial values are analyzed. The chaotic circuit has an infinite number of equilibrium points with multi-stability and exhibits coexisting bifurcations and coexisting attractors. Finally, the fractional order memristor-based chaotic circuit is verified by circuit simulations and DSP experiments.

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Complex Canard Explosion in a Fractional-Order FitzHugh–Nagumo Model

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127419501116 ◽

2019 ◽

Vol 29 (08) ◽

pp. 1950111 ◽

Cited By ~ 2

Author(s):

Mohammed-Salah Abdelouahab ◽

René Lozi ◽

Guanrong Chen

Keyword(s):

Dynamical System ◽

Fractional Order ◽

Complex Dynamics ◽

Search Algorithm ◽

Two Dimensional ◽

Mixed Mode Oscillations ◽

The Stability ◽

Two Parameters ◽

Complex Phenomena ◽

Autonomous Dynamical System

This article investigates the complex phenomena of canard explosion with mixed-mode oscillations, observed from a fractional-order FitzHugh–Nagumo (FFHN) model. To rigorously analyze the dynamics of the FFHN model, a new mathematical notion, referred to as Hopf-like bifurcation (HLB), is introduced. HLB provides a precise definition for the change between a fixed point and an [Formula: see text]-asymptotically [Formula: see text]-periodic solution of the fractional-order dynamical system, as well as the stability of the FFHN model and the appearance of the HLB. The existence of canard oscillations in the neighborhoods of such HLB points are numerically investigated. Using a new algorithm, referred to as the global-local canard explosion search algorithm, the appearance of various patterns of solutions is revealed, with an increasing number of small-amplitude oscillations when two key parameters of the FFHN model are varied. The numbers of such oscillations versus the two parameters, respectively, are perfectly fitted using exponential functions. Finally, it is conjectured that chaos could occur in a two-dimensional fractional-order autonomous dynamical system, with the fractional order close to one. After all, the article demonstrates that the FFHN model is a very simple two-dimensional model with an incredible ability to present the complex dynamics of neurons.

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Dynamic Analysis and Circuit Design of a Novel Hyperchaotic System with Fractional-Order Terms

Complexity ◽

10.1155/2017/3273408 ◽

2017 ◽

Vol 2017 ◽

pp. 1-10 ◽

Cited By ~ 21

Author(s):

Abir Lassoued ◽

Olfa Boubaker

Keyword(s):

Dynamic Analysis ◽

Circuit Design ◽

Fractional Order ◽

Potential Application ◽

Complex Dynamics ◽

Equilibrium Points ◽

Hyperchaotic System ◽

Software Simulation ◽

Simulation Results ◽

Hyperchaotic Systems

A novel hyperchaotic system with fractional-order (FO) terms is designed. Its highly complex dynamics are investigated in terms of equilibrium points, Lyapunov spectrum, and attractor forms. It will be shown that the proposed system exhibits larger Lyapunov exponents than related hyperchaotic systems. Finally, to enhance its potential application, a related circuit is designed by using the MultiSIM Software. Simulation results verify the effectiveness of the suggested circuit.

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BIFURCATIONS AND CHAOS IN FRACTIONAL-ORDER SIMPLIFIED LORENZ SYSTEM

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127410026411 ◽

2010 ◽

Vol 20 (04) ◽

pp. 1209-1219 ◽

Cited By ~ 68

Author(s):

KEHUI SUN ◽

XIA WANG ◽

J. C. SPROTT

Keyword(s):

Fractional Order ◽

Lorenz System ◽

Complex Dynamics ◽

Phase Portraits ◽

Largest Lyapunov Exponent ◽

Fractional Order Systems ◽

Wide Range ◽

The Time Domain ◽

Fractional Orders ◽

Simplified Lorenz System

The dynamics of fractional-order systems have attracted increasing attention in recent years. In this paper, we numerically study the bifurcations and chaotic behaviors in the fractional-order simplified Lorenz system using the time-domain scheme. Chaos does exist in this system for a wide range of fractional orders, both less than and greater than three. Complex dynamics with interesting characteristics are presented by means of phase portraits, bifurcation diagrams and the largest Lyapunov exponent. Both the system parameter and the fractional order can be taken as bifurcation parameters, and the range of existing chaos is different for different parameters. The lowest order we found for this system to yield chaos is 2.62.

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Dynamical analysis of fractional-order differentiated Cournot triopoly game

Journal of Vibration and Control ◽

10.1177/1077546319896125 ◽

2020 ◽

Vol 26 (15-16) ◽

pp. 1367-1380

Author(s):

Abdulrahman Al-khedhairi

Keyword(s):

Stability Analysis ◽

Fractional Order ◽

Complex Dynamics ◽

Equilibrium Points ◽

Sufficient Conditions ◽

Dynamical Analysis ◽

Periodic Forcing ◽

Nash Equilibrium Point ◽

Dynamical Behaviors ◽

Theoretical Results

The objective of the article is to study the dynamics of the proposed fractional-order Cournot triopoly game. Sufficient conditions for the existence and uniqueness of the triopoly game solution are obtained. Stability analysis of equilibrium points of the fractional-order game is also discussed. The conditions for the presence of Nash equilibrium point along with its global stability analysis are studied. The interesting dynamical behaviors of the arbitrary-order Cournot triopoly game are discussed. Moreover, the effects of seasonal periodic forcing on the game’s behaviors are examined. The 0–1 test is used to distinguish between regular and irregular dynamics of system behaviors. Numerical analysis is used to verify the theoretical results that are obtained, and revealed that the nonautonomous fractional-order model induces more complicated dynamics in the Cournot triopoly game behavior and the seasonally forced game exhibits more complex dynamics than the unforced one.

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Complex Dynamics of the Fractional-Order Rössler System and Its Tracking Synchronization Control

Complexity ◽

10.1155/2018/4019749 ◽

2018 ◽

Vol 2018 ◽

pp. 1-13 ◽

Cited By ~ 2

Author(s):

Huihai Wang ◽

Shaobo He ◽

Kehui Sun

Keyword(s):

Fractional Order ◽

Chaotic Systems ◽

Complex Dynamics ◽

Adomian Decomposition Method ◽

Adomian Decomposition ◽

Chaotic Signals ◽

Rossler System ◽

Master System ◽

Rössler System ◽

Tracking Synchronization

Numerical analysis of fractional-order chaotic systems is a hot topic of recent years. The fractional-order Rössler system is solved by a fast discrete iteration which is obtained from the Adomian decomposition method (ADM) and it is implemented on the DSP board. Complex dynamics of the fractional-order chaotic system are analyzed by means of Lyapunov exponent spectra, bifurcation diagrams, and phase diagrams. It shows that the system has rich dynamics with system parameters and the derivative order q. Moreover, tracking synchronization controllers are theoretically designed and numerically investigated. The system can track different signals including chaotic signals from the fractional-order master system and constant signals. It lays a foundation for the application of the fractional-order Rössler system.

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Firing activities of a fractional-order FitzHugh-Rinzel bursting neuron model and its coupled dynamics

Scientific Reports ◽

10.1038/s41598-019-52061-4 ◽

2019 ◽

Vol 9 (1) ◽

Cited By ~ 4

Author(s):

Argha Mondal ◽

Sanjeev Kumar Sharma ◽

Ranjit Kumar Upadhyay ◽

Arnab Mondal

Keyword(s):

Fractional Order ◽

Complex Dynamics ◽

Firing Frequency ◽

Order System ◽

Neuron Models ◽

Order Model ◽

Constant Input ◽

Wide Range ◽

Excitable Systems ◽

Firing Properties

Abstract Fractional-order dynamics of excitable systems can be physically described as a memory dependent phenomenon. It can produce diverse and fascinating oscillatory patterns for certain types of neuron models. To address these characteristics, we consider a nonlinear fast-slow FitzHugh-Rinzel (FH-R) model that exhibits elliptic bursting at a fixed set of parameters with a constant input current. The generalization of this classical order model provides a wide range of neuronal responses (regular spiking, fast-spiking, bursting, mixed-mode oscillations, etc.) in understanding the single neuron dynamics. So far, it is not completely understood to what extent the fractional-order dynamics may redesign the firing properties of excitable systems. We investigate how the classical order system changes its complex dynamics and how the bursting changes to different oscillations with stability and bifurcation analysis depending on the fractional exponent (0 < α ≤ 1). This occurs due to the memory trace of the fractional-order dynamics. The firing frequency of the fractional-order FH-R model is less than the classical order model, although the first spike latency exists there. Further, we investigate the responses of coupled FH-R neurons with small coupling strengths that synchronize at specific fractional-orders. The interesting dynamical characteristics suggest various neurocomputational features that can be induced in this fractional-order system which enriches the functional neuronal mechanisms.

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Fractional Hyperchaotic Telecommunication Systems: A New Paradigm

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4023165 ◽

2013 ◽

Vol 8 (3) ◽

Cited By ~ 15

Author(s):

Abolhassan Razminia ◽

Dumitru Baleanu

Keyword(s):

Fractional Order ◽

Data Transmission ◽

Secure Communication ◽

Complex Dynamics ◽

Order System ◽

Fractional Order Systems ◽

New Paradigm ◽

Telecommunication System ◽

Telecommunication Systems ◽

Synchronization Scheme

The dynamics of hyperchaotic and fractional-order systems have increasing attracted attention in recent years. In this paper, we mix two complex dynamics to construct a new telecommunication system. Using a hyperchaotic fractional order system, we propose a novel synchronization scheme between receiver and transmitter which increases the security of data transmission and communication. Indeed, this is first work that can open a new way in secure communication system.

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Complex dynamics behavior analysis of a new chaotic system based on fractional-order memristor

Journal of Physics Conference Series ◽

10.1088/1742-6596/1861/1/012114 ◽

2021 ◽

Vol 1861 (1) ◽

pp. 012114

Author(s):

Yongwei Qi ◽

Chaojun Wu ◽

Qi Zhang ◽

Kai Yan ◽

Haohan Wang

Keyword(s):

Behavior Analysis ◽

Fractional Order ◽

Chaotic System ◽

Complex Dynamics ◽

New Chaotic System

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Chaotic Behavior Analysis of a New Incommensurate Fractional-Order Hopfield Neural Network System

Complexity ◽

10.1155/2021/3394666 ◽

2021 ◽

Vol 2021 ◽

pp. 1-11

Author(s):

Nadjette Debbouche ◽

Adel Ouannas ◽

Iqbal M. Batiha ◽

Giuseppe Grassi ◽

Mohammed K. A. Kaabar ◽

...

Keyword(s):

Neural Network ◽

Fractional Order ◽

Complex Dynamics ◽

Chaotic Behavior ◽

Hopfield Neural Network ◽

Phase Portraits ◽

Chaotic Attractors ◽

Neural Network System ◽

The Stability ◽

Arbitrary Location

This study intends to examine different dynamics of the chaotic incommensurate fractional-order Hopfield neural network model. The stability of the proposed incommensurate-order model is analyzed numerically by continuously varying the values of the fractional-order derivative and the values of the system parameters. It turned out that the formulated system using the Caputo differential operator exhibits many rich complex dynamics, including symmetry, bistability, and coexisting chaotic attractors. On the other hand, it has been detected that by adapting the corresponding controlled constants, such systems possess the so-called offset boosting of three variables. Besides, the resultant periodic and chaotic attractors can be scattered in several forms, including 1D line, 2D lattice, and 3D grid, and even in an arbitrary location of the phase space. Several numerical simulations are implemented, and the obtained findings are illustrated through constructing bifurcation diagrams, computing Lyapunov exponents, calculating Lyapunov dimensions, and sketching the phase portraits in 2D and 3D projections.

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