Multi-Scroll Attractors with Hyperchaotic Behavior Using Fractional-Order Systems

2021 ◽  
Author(s):  
Kenan Altun
Keyword(s):  
Fractional Order ◽  
Error Function ◽  
Equilibrium Points ◽  
Test Methods ◽  
Systematic Design ◽  
Fractional Order Systems ◽  
Numerical Investigations ◽  
Dynamical Behaviors ◽  
Chaotic Signals ◽  
Methods Numerical

In this paper, a systematic design is proposed to generate multi-scroll attractors with hyperchaotic behavior using fractional-order systems, in which switched SC-CNN is triggered with error function. Sprott Systems Case H is reconstructed with fractional-order switched SC-CNN system. Herein, the goal is to increase the complexity of chaotic signals, hence providing safer and reliable communication by generating multi-scroll attractors with hyperchaotic behavior using fractional-order systems. Theoretical analysis of the proposed system’s dynamical behaviors is scrutinized, while numerical investigations are carried out with equilibrium points, Lyapunov exponent, bifurcation diagrams, Poincaré mapping and 0/1 test methods. Numerical results are validated through simulations and on an FPAA platform.

scholarly journals Theoretical Analysis and Circuit Verification for Fractional-Order Chaotic Behavior in a New Hyperchaotic System

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Ling Liu ◽  
Chongxin Liu
Keyword(s):  
Fractional Order ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Circuit Diagram ◽  
Phase Portraits ◽  
Hyperchaotic System ◽  
Dynamical Behaviors ◽  
Mapping Parameter ◽  
New Hyperchaotic System ◽  
Observation Results

A novel nonlinear four-dimensional hyperchaotic system and its fractional-order form are presented. Some dynamical behaviors of this system are further investigated, including Poincaré mapping, parameter phase portraits, equilibrium points, bifurcations, and calculated Lyapunov exponents. A simple fourth-channel block circuit diagram is designed for generating strange attractors of this dynamical system. Specifically, a novel network module fractance is introduced to achieve fractional-order circuit diagram for hardware implementation of the fractional attractors of this nonlinear hyperchaotic system with order as low as 0.9. Observation results have been observed by using oscilloscope which demonstrate that the fractional-order nonlinear hyperchaotic attractors exist indeed in this new system.

Dynamical analysis of fractional-order differentiated Cournot triopoly game

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.

scholarly journals Control and Synchronization of the Fractional-Order Lorenz Chaotic System via Fractional-Order Derivative

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Ping Zhou ◽  
Rui Ding
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Stability Theory ◽  
Chaos Synchronization ◽  
Equilibrium Points ◽  
Order Derivative ◽  
Fractional Order Systems ◽  
Fractional Order Derivative ◽  
Lorenz Chaotic System ◽  
Control And Synchronization

The unstable equilibrium points of the fractional-order Lorenz chaotic system can be controlled via fractional-order derivative, and chaos synchronization for the fractional-order Lorenz chaotic system can be achieved via fractional-order derivative. The control and synchronization technique, based on stability theory of fractional-order systems, is simple and theoretically rigorous. The numerical simulations demonstrate the validity and feasibility of the proposed method.

scholarly journals Chaotic Discrete Fractional-Order Food Chain Model and Hybrid Image Encryption Scheme Application

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 161
Author(s):  
Sameh Askar ◽  
Abdulrahman Al-khedhairi ◽  
Amr Elsonbaty ◽  
Abdelalim Elsadany
Keyword(s):  
Fractional Order ◽  
Food Chain ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
High Sensitivity ◽  
Phase Portraits ◽  
Chain Model ◽  
Encryption Scheme ◽  
Dynamical Behaviors ◽  
Food Chain Model

Using the discrete fractional calculus, a novel discrete fractional-order food chain model for the case of strong pressure on preys map is proposed. Dynamical behaviors of the model involving stability analysis of its equilibrium points, bifurcation diagrams and phase portraits are investigated. It is demonstrated that the model can exhibit a variety of dynamical behaviors including stable steady states, periodic and quasiperiodic dynamics. Then, a hybrid encryption scheme based on chaotic behavior of the model along with elliptic curve key exchange scheme is proposed for colored plain images. The hybrid scheme combines the characteristics of noise-like chaotic dynamics of the map, including high sensitivity to values of parameters, with the advantages of reliable elliptic curves-based encryption systems. Security analysis assures the efficiency of the proposed algorithm and validates its robustness and efficiency against possible types of attacks.

scholarly journals Chaos in a Financial System with Fractional Order and Its Control via Sliding Mode

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Paul Yaovi Dousseh ◽  
Cyrille Ainamon ◽  
Clément Hodévèwan Miwadinou ◽  
Adjimon Vincent Monwanou ◽  
Jean Bio Chabi Orou
Keyword(s):  
Fractional Order ◽  
Chaos Control ◽  
Sliding Mode ◽  
Financial System ◽  
Period Doubling ◽  
Fractional Order Systems ◽  
Uncertain Dynamics ◽  
Dynamical Behaviors ◽  
Order Case ◽  
Route To Chaos

In this paper, the dynamical behaviors and chaos control of a fractional-order financial system are discussed. The lowest fractional order found from which the system generates chaos is 2.49 for the commensurate order case and 2.13 for the incommensurate order case. Also, period-doubling route to chaos was found in this system. The results of this study were validated by the existence of a positive Lyapunov exponent. Besides, in order to control chaos in this fractional-order financial system with uncertain dynamics, a sliding mode controller is derived. The proposed controller stabilizes the commensurate and incommensurate fractional-order systems. Numerical simulations are carried out to verify the analytical results.

scholarly journals Dynamical Behaviors of a Fractional-Order Three Dimensional Prey-Predator Model

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Noor S. Sh. Barhoom ◽  
Sadiq Al-Nassir
Keyword(s):  
Fractional Order ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Predator Species ◽  
Dynamical Behaviors ◽  
Proposed Model ◽  
Constant Rate ◽  
Predator Model ◽  
Theoretical Results

In this paper, the dynamical behavior of a three-dimensional fractional-order prey-predator model is investigated with Holling type III functional response and constant rate harvesting. It is assumed that the middle predator species consumes only the prey species, and the top predator species consumes only the middle predator species. We also prove the boundedness, the non-negativity, the uniqueness, and the existence of the solutions of the proposed model. Then, all possible equilibria are determined, and the dynamical behaviors of the proposed model around the equilibrium points are investigated. Finally, numerical simulations results are presented to confirm the theoretical results and to give a better understanding of the dynamics of our proposed model.

A Series of New Chaotic Attractors via Switched Linear Integer Order and Fractional Order Differential Equations

2015 ◽  
Vol 25 (01) ◽  
pp. 1550008 ◽  
Author(s):  
Fei Xu ◽  
Ross Cressman ◽  
Xiao-Bao Shu ◽  
Xinzhi Liu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Order ◽  
Order Differential Equation ◽  
Chaotic Attractors ◽  
Fractional Order Systems ◽  
Fractional Order Differential Equation ◽  
Integer Order ◽  
Fractional Order Differential Equations ◽  
Dynamical Behaviors

This article investigates the design of a series of new chaotic attractors. A switching control with different switching surfaces is designed to link two systems of linear integer order differential equations. Under such control, the linked systems have rich dynamical behaviors such as chaos. We also investigate the dynamical behaviors of the corresponding linear fractional order differential equation systems with switching controls. It is shown that such fractional order systems have chaotic behaviors as well.

Control Fractional-Order Continuous Chaotic System via a Simple Fractional-Order Controller

2013 ◽  
Vol 336-338 ◽  
pp. 770-773
Author(s):  
Dong Zhang ◽  
Shou Liang Yang
Keyword(s):  
Numerical Simulations ◽  
Fractional Order ◽  
Equilibrium Points ◽  
Stability Theorem ◽  
Asymptotically Stable ◽  
Fractional Order Systems ◽  
Control Scheme ◽  
The Stability ◽  
Fractional Order Controller ◽  
Order Continuous

A universal fractional-order controller is proposed to asymptotically stable the unstable equilibrium points and the nonequilibrium points of continuous fractional-order chaos systems. The simple fractional-order controller is obtained based on the stability theorem of nonlinear fractional-order systems. The control scheme is simple and theoretically rigorous. The numerical simulations demonstrate the validity and feasibility of the proposed fractional-order controller.

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.

scholarly journals On Class of Fractional-Order Chaotic or Hyperchaotic Systems in the Context of the Caputo Fractional-Order Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Ndolane Sene ◽  
Ameth Ndiaye
Keyword(s):  
Lyapunov Exponents ◽  
Fractional Order ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Phase Portraits ◽  
Order Derivative ◽  
Order System ◽  
Fractional Order Systems ◽  
Fractional Order Derivative ◽  
The Stability

In this paper, we consider a class of fractional-order systems described by the Caputo derivative. The behaviors of the dynamics of this particular class of fractional-order systems will be proposed and experienced by a numerical scheme to obtain the phase portraits. Before that, we will provide the conditions under which the considered fractional-order system’s solution exists and is unique. The fractional-order impact will be analyzed, and the advantages of the fractional-order derivatives in modeling chaotic systems will be discussed. How the parameters of the model influence the considered fractional-order system will be studied using the Lyapunov exponents. The topological changes of the systems and the detection of the chaotic and hyperchaotic behaviors at the assumed initial conditions and the considered fractional-order systems will also be investigated using the Lyapunov exponents. The investigations related to the Lyapunov exponents in the context of the fractional-order derivative will be the main novelty of this paper. The stability analysis of the model’s equilibrium points has been focused in terms of the Matignon criterion.

Sign in / Sign up

Export Citation Format

Share Document