scholarly journals Dynamic Behaviors and the Equivalent Realization of a Novel Fractional-Order Memristor-Based Chaotic Circuit

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Ningning Yang ◽  
Cheng Xu ◽  
Chaojun Wu ◽  
Rong Jia ◽  
Chongxin Liu
Keyword(s):  
Numerical Simulations ◽  
Dynamic Behavior ◽  
Fractional Order ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Great Influence ◽  
Equivalent Circuits ◽  
Chaotic Circuit ◽  
Undetermined Coefficient ◽  
The Stability

This paper proposed a novel fractional-order memristor-based chaotic circuit. A memristive diode bridge cascaded with a fractional-order RL filter constitutes the generalized fractional-order memristor. The mathematical model of the proposed fractional-order chaotic circuit is established by extending the nonlinear capacitor and inductor in the memristive chaotic circuit to the fractional order. Detailed theoretical analysis and numerical simulations are carried out on the dynamic behavior of the proposed circuit by investigating the stability of equilibrium points and the influence of circuit parameters on bifurcations. The results show that the order of the fractional-order circuit has a great influence on the dynamical behavior of the system. The system may exhibit complicated nonlinear dynamic behavior such as bifurcation and chaos with the change of the order. The equivalent circuits of the fractional-order inductor and capacitor are also given in the paper, and the parameters of the equivalent circuits are solved by an undetermined coefficient method. Circuit simulations of the equivalent fractional-order memristive chaotic circuit are carried out in order to validate the correctness of numerical simulations and the practicability of using the integer-order equivalent circuit to substitute the fractional-order element.

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.

Optimal Synchronization of a Memristive Chaotic Circuit

2016 ◽  
Vol 26 (06) ◽  
pp. 1650093 ◽  
Author(s):  
Michaux Kountchou ◽  
Patrick Louodop ◽  
Samuel Bowong ◽  
Hilaire Fotsin ◽  
Jurgen Kurths
Keyword(s):  
Numerical Simulations ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Analog Circuit ◽  
Electronic Circuit ◽  
Dynamical Behavior ◽  
Circuit Implementation ◽  
Chaotic Circuit ◽  
Dynamical Properties ◽  
Analog Circuit Implementation

This paper deals with the problem of optimal synchronization of two identical memristive chaotic systems. We first study some basic dynamical properties and behaviors of a memristor oscillator with a simple topology. An electronic circuit (analog simulator) is proposed to investigate the dynamical behavior of the system. An optimal synchronization strategy based on the controllability functions method with a mixed cost functional is investigated. A finite horizon is explicitly computed such that the chaos synchronization is achieved at an established time. Numerical simulations are presented to verify the effectiveness of the proposed synchronization strategy. Pspice analog circuit implementation of the complete master-slave-controller systems is also presented to show the feasibility of the proposed scheme.

scholarly journals Theory and applications of new fractional-order chaotic system under Caputo operator

2021 ◽  
Vol 12 (1) ◽  
pp. 20-38
Author(s):  
Ndolane Sene
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Caputo Derivative ◽  
Phase Portraits ◽  
Chaotic Circuit ◽  
Chaotic Model ◽  
The Stability ◽  
The Impact ◽  
Schematic Circuit ◽  
Good Agreement

This paper introduces the properties of a fractional-order chaotic system described by the Caputo derivative. The impact of the fractional-order derivative has been focused on. The phase portraits in different orders are obtained with the aids of the proposed numerical discretization, including the discretization of the Riemann-Liouville fractional integral. The stability analysis has been used to help us to delimit the chaotic region. In other words, the region where the order of the Caputo derivative involves and where the presented system in this paper is chaotic. The nature of the chaos has been established using the Lyapunov exponents in the fractional context. The schematic circuit of the proposed fractional-order chaotic system has been presented and simulated in via Mutltisim. The results obtained via Multisim simulation of the chaotic circuit are in good agreement with the results with Matlab simulations. That provided the fractional operators can be applied in real- worlds applications as modeling electrical circuits. The presence of coexisting attractors for particular values of the parameters of the presented fractional-order chaotic model has been studied.

scholarly journals A Nonvolatile Fractional Order Memristor Model and its Complex Dynamics

Entropy ◽  
2019 ◽  
Vol 21 (10) ◽  
pp. 955 ◽  
Author(s):  
Wu ◽  
Wang ◽  
Iu ◽  
Shen ◽  
Zhou
Keyword(s):  
Fractional Order ◽  
Complex Dynamics ◽  
Equilibrium Points ◽  
Stable State ◽  
Electrical Characteristics ◽  
Chaotic Circuit ◽  
Stable States ◽  
Road Map ◽  
Chaotic Circuits ◽  
Integral Order

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.

scholarly journals A Fractional-Order Discrete Noninvertible Map of Cubic Type: Dynamics, Control, and Synchronization

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-21
Author(s):  
Xiaojun Liu ◽  
Ling Hong ◽  
Lixin Yang ◽  
Dafeng Tang
Keyword(s):  
Fractional Order ◽  
Initial Conditions ◽  
Dimensional Space ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
State Variables ◽  
Discrete Maps ◽  
The Stability ◽  
Control And Synchronization ◽  
Simultaneous Variation

In this paper, a new fractional-order discrete noninvertible map of cubic type is presented. Firstly, the stability of the equilibrium points for the map is examined. Secondly, the dynamics of the map with two different initial conditions is studied by numerical simulation when a parameter or a derivative order is varied. A series of attractors are displayed in various forms of periodic and chaotic ones. Furthermore, bifurcations with the simultaneous variation of both a parameter and the order are also analyzed in the three-dimensional space. Interior crises are found in the map as a parameter or an order varies. Thirdly, based on the stability theory of fractional-order discrete maps, a stabilization controller is proposed to control the chaos of the map and the asymptotic convergence of the state variables is determined. Finally, the synchronization between the proposed map and a fractional-order discrete Loren map is investigated. Numerical simulations are used to verify the effectiveness of the designed synchronization controllers.

Dynamical Analysis and Circuit Simulation of a New Fractional-Order Hyperchaotic System and Its Discretization

2016 ◽  
Vol 26 (13) ◽  
pp. 1650222 ◽  
Author(s):  
A. M. A. El-Sayed ◽  
A. Elsonbaty ◽  
A. A. Elsadany ◽  
A. E. Matouk
Keyword(s):  
Fractional Order ◽  
Initial Conditions ◽  
Circuit Simulation ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Control Technique ◽  
Order System ◽  
Qualitative Behavior ◽  
Hyperchaotic System

This paper presents an analytical framework to investigate the dynamical behavior of a new fractional-order hyperchaotic circuit system. A sufficient condition for existence, uniqueness and continuous dependence on initial conditions of the solution of the proposed system is derived. The local stability of all the system’s equilibrium points are discussed using fractional Routh–Hurwitz test. Then the analytical conditions for the existence of a pitchfork bifurcation in this system with fractional-order parameter less than 1/3 are provided. Conditions for the existence of Hopf bifurcation in this system are also investigated. The dynamics of discretized form of our fractional-order hyperchaotic system are explored. Chaos control is also achieved in discretized system using delay feedback control technique. The numerical simulation are presented to confirm our theoretical analysis via phase portraits, bifurcation diagrams and Lyapunov exponents. A text encryption algorithm is presented based on the proposed fractional-order system. The results show that the new system exhibits a rich variety of dynamical behaviors such as limit cycles, chaos and transient phenomena where fractional-order derivative represents a key parameter in determining system qualitative behavior.

Self-Excited and Hidden Attractors in an Autonomous Josephson Jerk Oscillator: Analysis and Its Application to Text Encryption

2019 ◽  
Vol 14 (7) ◽  
Author(s):  
Sifeu Takougang Kingni ◽  
Gaetan Fautso Kuiate ◽  
Victor Kamdoum Tamba ◽  
Viet-Thanh Pham ◽  
Duy Vo Hoang
Keyword(s):  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Projective Synchronization ◽  
Chaotic Attractors ◽  
Generalized Function ◽  
Function Projective Synchronization ◽  
Dc Bias ◽  
The Stability ◽  
Security Device ◽  
High Level

By converting the resistive capacitive shunted junction model to a jerk oscillator, an autonomous chaotic Josephson jerk oscillator which can belong to oscillators with hidden and self-excited attractors is designed. The proposed autonomous Josephson jerk oscillator has two or no equilibrium points depending on DC bias current. The stability analysis of the two equilibrium points shows that one of the equilibrium points is unstable while for the other equilibrium point, the existence of a Hopf bifurcation is established. The dynamical behavior of autonomous Josephson jerk oscillator is analyzed by using standard tools of nonlinear analysis. For a suitable choice of the parameters, an autonomous Josephson jerk oscillator can generate antimonotonicity, periodic oscillations, self-excited chaotic attractors, hidden chaotic attractors, hidden chaotic bubble attractors, and coexistence between periodic and chaotic self-excited attractors. Finally, a text cryptographic encryption scheme with the help of generalized function projective synchronization of the proposed autonomous Josephson jerk oscillators in hidden chaotic attractor regime is illustrated through a numerical example, showing that a high-level security device can be produced using this system.

Chaotic Oscillator Based on Fractional Order Memcapacitor

2019 ◽  
Vol 28 (14) ◽  
pp. 1950239 ◽  
Author(s):  
Akif Akgul
Keyword(s):  
Mathematical Model ◽  
Lyapunov Exponents ◽  
Fractional Order ◽  
Nonlinear Oscillator ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Chaotic Oscillator ◽  
State Equations ◽  
Chaotic Oscillators ◽  
Eigen Values

Many literatures have discussed fractional order memristor and memcapacitor-based chaotic oscillators but the entire oscillator model is considered to be of fractional order. My interest is to propose a nonlinear oscillator with considering only the memcapacitor element of fractional order. Hence, I propose a fractional order memcapacitor (FMC)-based novel chaotic oscillator. The complete mathematical model for the proposed oscillator is derived and presented in this paper. The dimensionless state equations are then analyzed by using the equilibrium points and their stability, Eigen values, Kaplan–Yorke dimensions and Lyapunov exponents. To understand the complete dynamical behavior, bifurcation graphs are obtained and presented. Finally, the proposed fractional memcapacitor oscillator is implemented by using the shelf components.

A GLOBAL PERIOD-1 MOTION OF A PERIODICALLY EXCITED, PIECEWISE-LINEAR SYSTEM

2005 ◽  
Vol 15 (06) ◽  
pp. 1945-1957 ◽  
Author(s):  
SANTHOSH MENON ◽  
ALBERT C. J. LUO
Keyword(s):  
Linear System ◽  
Numerical Simulations ◽  
Piecewise Linear ◽  
Dynamical Behavior ◽  
Analytical Prediction ◽  
Algebraic Equations ◽  
Piecewise Linear System ◽  
Nonlinear Algebraic Equations ◽  
The Stability ◽  
Nonsmooth Dynamical Systems

The period-1 motion of a piecewise-linear system under a periodic excitation is predicted analytically through the Poincaré mapping and the corresponding mapping sections formed by the switch planes pertaining to the two constraints. The mapping relationship generates a set of nonlinear algebraic equations from which the period-1 motion is determined analytically. The stability and bifurcation of the period-1 motion are determined, and numerical simulations are carried out for confirmation of the analytical prediction of period-1 motion. An unsymmetrical stable period-1 motion is observed. This investigation helps us understand the dynamical behavior of period-1 motion in the piecewise-linear system and more efficiently obtain other periodic motions and chaos through numerical simulations. The similar methodology presented in this paper can be used for other nonsmooth dynamical systems.

Investigation of Synchronization for a Fractional-Order Delayed System

2014 ◽  
Vol 687-691 ◽  
pp. 447-450 ◽  
Author(s):  
Hong Gang Dang ◽  
Wan Sheng He ◽  
Xiao Ya Yang
Keyword(s):  
Numerical Simulations ◽  
Fractional Order ◽  
Stability Theory ◽  
Fractional Order Systems ◽  
Delayed System ◽  
The Stability

In this paper, synchronization of a fractional-order delayed system is studied. Based on the stability theory of fractional-order systems, by designing appropriate controllers, the synchronization for the proposed system is achieved. Numerical simulations show the effectiveness of the proposed scheme.

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