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.

scholarly journals Optimizing the Kaplan–Yorke Dimension of Chaotic Oscillators Applying DE and PSO

Technologies ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 38 ◽  
Author(s):  
Alejandro Silva-Juarez ◽  
Gustavo Rodriguez-Gomez ◽  
Luis Gerardo de la Fraga ◽  
Omar Guillen-Fernandez ◽  
Esteban Tlelo-Cuautle
Keyword(s):  
Mathematical Model ◽  
Time Series ◽  
Color Image ◽  
Equilibrium Points ◽  
Computing Time ◽  
Chaotic Oscillator ◽  
State Variables ◽  
Huge Number ◽  
Chaotic Oscillators ◽  
Swarm Optimization

When a new chaotic oscillator is introduced, it must accomplish characteristics like guaranteeing the existence of a positive Lyapunov exponent and a high Kaplan–Yorke dimension. In some cases, the coefficients of a mathematical model can be varied to increase the values of those characteristics but it is not a trivial task because a very huge number of combinations arise and the required computing time can be unreachable. In this manner, we introduced the optimization of the Kaplan–Yorke dimension of chaotic oscillators by applying metaheuristics, e.g., differential evolution (DE) and particle swarm optimization (PSO) algorithms. We showed the equilibrium points and eigenvalues of three chaotic oscillators that are simulated applying ODE45, and the Kaplan–Yorke dimension was evaluated by Wolf’s method. The chaotic time series of the state variables associated to the highest Kaplan–Yorke dimension provided by DE and PSO are used to encrypt a color image to demonstrate that they are useful in implementing a secure chaotic communication system. Finally, the very low correlation between the chaotic channel and the original color image confirmed the usefulness of optimizing Kaplan–Yorke dimension for cryptographic applications.

scholarly journals MATHEMATICAL MODEL OF THREE SPECIES FOOD CHAIN INTERACTION WITH MIXED FUNCTIONAL RESPONSE

2012 ◽  
Vol 09 ◽  
pp. 334-340 ◽  
Author(s):  
MADA SANJAYA WS ◽  
ISMAIL BIN MOHD ◽  
MUSTAFA MAMAT ◽  
ZABIDIN SALLEH
Keyword(s):  
Mathematical Model ◽  
Functional Response ◽  
Food Chain ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Bifurcation Points ◽  
Dynamical Behaviors ◽  
Practical Life ◽  
Parameter Values ◽  
Chain Interaction

In this paper, we study mathematical model of ecology with a tritrophic food chain composed of a classical Lotka-Volterra functional response for prey and predator, and a Holling type-III functional response for predator and super predator. There are two equilibrium points of the system. In the parameter space, there are passages from instability to stability, which are called Hopf bifurcation points. For the first equilibrium point, it is possible to find bifurcation points analytically and to prove that the system has periodic solutions around these points. Furthermore the dynamical behaviors of this model are investigated. Models for biologically reasonable parameter values, exhibits stable, unstable periodic and limit cycles. The dynamical behavior is found to be very sensitive to parameter values as well as the parameters of the practical life. Computer simulations are carried out to explain the analytical findings.

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.

Fractal-fractional order dynamical behavior of an HIV/AIDS epidemic mathematical model

2021 ◽  
Vol 136 (1) ◽  
Author(s):  
Zeeshan Ali ◽  
Faranak Rabiei ◽  
Kamal Shah ◽  
Touraj Khodadadi
Keyword(s):  
Mathematical Model ◽  
Fractional Order ◽  
Dynamical Behavior ◽  
Aids Epidemic ◽  
Hiv Aids

Dynamic analysis of a fractional order system

2015 ◽  
Vol 08 (06) ◽  
pp. 1550079
Author(s):  
M. Javidi ◽  
N. Nyamoradi
Keyword(s):  
Stability Analysis ◽  
Fractional Derivative ◽  
Dynamic Analysis ◽  
Fractional Order ◽  
Local Stability ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Order System ◽  
Stability Conditions ◽  
Hurwitz Stability

In this paper, we investigate the dynamical behavior of a fractional order phytoplankton–zooplankton system. In this paper, stability analysis of the phytoplankton–zooplankton model (PZM) is studied by using the fractional Routh–Hurwitz stability conditions. We have studied the local stability of the equilibrium points of PZM. We applied an efficient numerical method based on converting the fractional derivative to integer derivative to solve the PZM.

scholarly journals Determining the Lyapunov Spectrum of Continuous-Time 1D and 2D Multiscroll Chaotic Oscillators via the Solution ofm-PWL Variational Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Jesus Manuel Munoz-Pacheco ◽  
Luz del Carmen Gómez-Pavón ◽  
Olga Guadalupe Félix-Beltrán ◽  
Arnulfo Luis-Ramos
Keyword(s):  
Lyapunov Exponents ◽  
Dynamic Range ◽  
Piecewise Linear ◽  
Lyapunov Spectrum ◽  
System Level ◽  
Design Parameters ◽  
Chaotic Oscillator ◽  
Chaotic Oscillators ◽  
Variational Equations ◽  
The Impact

An algorithm to compute the Lyapunov exponents of piecewise linear function-based multidirectional multiscroll chaotic oscillators is reported. Based on themregions in the piecewise linear functions, the suggested algorithm determines the individual expansion rate of Lyapunov exponents fromm-piecewise linear variational equations and their associatedm-Jacobian matrices whose entries remain constant during all computation cycles. Additionally, by considering OpAmp-based chaotic oscillators, we study the impact of two analog design procedures on the magnitude of Lyapunov exponents. We focus on analyzing variations of both frequency bandwidth and voltage/current dynamic range of the chaotic signals at electronic system level. As a function of the design parameters, a renormalization factor is proposed to estimate correctly the Lyapunov spectrum. Numerical simulation results in a double-scroll type chaotic oscillator and complex chaotic oscillators generating multidirectional multiscroll chaotic attractors on phase space confirm the usefulness of the reported algorithm.

scholarly journals Qualitative Analysis of Class of Fractional-Order Chaotic System via Bifurcation and Lyapunov Exponents Notions

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Ndolane Sene
Keyword(s):  
Lyapunov Exponents ◽  
Fractional Order ◽  
Chaotic System ◽  
Numerical Scheme ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Phase Portraits ◽  
Fractional Chaotic System ◽  
The Stability ◽  
Predictor Corrector

This paper presents a modified chaotic system under the fractional operator with singularity. The aim of the present subject will be to focus on the influence of the new model’s parameters and its fractional order using the bifurcation diagrams and the Lyapunov exponents. The new fractional model will generate chaotic behaviors. The Lyapunov exponents’ theories in fractional context will be used for the characterization of the chaotic behaviors. In a fractional context, the phase portraits will be obtained with a predictor-corrector numerical scheme method. The details of the numerical scheme will be presented in this paper. The numerical scheme will be used to analyze all the properties addressed in this present paper. The Matignon criterion will also play a fundamental role in the local stability of the presented model’s equilibrium points. We will find a threshold under which the stability will be removed and the chaotic and hyperchaotic behaviors will be generated. An adaptative control will be proposed to correct the instability of the equilibrium points of the model. Sensitive to the initial conditions, we will analyze the influence of the initial conditions on our fractional chaotic system. The coexisting attractors will also be provided for illustrations of the influence of the initial conditions.

scholarly journals Fractional-Order Hidden Attractor Based on the Extended Liu System

2020 ◽  
Vol 2020 ◽  
pp. 1-22 ◽  
Author(s):  
Yaoyu Wang ◽  
Ling Liu ◽  
Xinshan Cai ◽  
Chongxin Liu ◽  
Yan Wang ◽  
...  
Keyword(s):  
Equilibrium Point ◽  
Fractional Order ◽  
Finite Time ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Chaotic Oscillator ◽  
Hidden Attractor ◽  
Fractional System ◽  
Line Equilibrium ◽  
Liu System

In this paper, a new commensurate fractional-order chaotic oscillator is presented. The mathematical model with a weak feedback term, which is named hypogenetic flow, is proposed based on the Liu system. And with changing the parameters of the system, the hidden attractor can have no equilibrium points or line equilibrium. What is more interesting is that under the occasion that no equilibrium point can be obtained, the phase trajectory can converge to a minimal field under the lead of some initial conditions, similar to the fixed point. We call it the virtual equilibrium point. On the other hand, when the value of parameters can produce an infinite number of equilibrium points, the line equilibrium points are nonhyperbolic. Moreover than that, there are coexistence attractors, which can present hyperchaos, chaos, period, and virtual equilibrium point. The dynamic characteristics of the system are analyzed, and the parameter estimation is also studied. Then, an electronic circuit implementation of the system is built, which shows the feasibility of the system. At last, for the fractional system with hidden attractors, the finite-time synchronization control of the system is carried out based on the finite-time stability theory of the fractional system. And the effectiveness of the controller is verified by numerical simulation.

EXPERIMENTAL VERIFICATION OF A FOUR-DIMENSIONAL CHUA'S SYSTEM AND ITS FRACTIONAL ORDER CHAOTIC ATTRACTORS

2009 ◽  
Vol 19 (08) ◽  
pp. 2473-2486 ◽  
Author(s):  
LING LIU ◽  
CHONGXIN LIU ◽  
YANBIN ZHANG
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Analog Circuit ◽  
Equilibrium Points ◽  
Phase Portraits ◽  
Chaotic Oscillator ◽  
Oscillation Circuit ◽  
Mapping Parameter ◽  
Time Autonomous ◽  
First Time

This paper introduces a modified Chua's system which is a smooth four-dimensional continuous-time autonomous chaotic system with a cubic nonlinearity. Some dynamical behaviors of this 4-D Chua's system are further investigated by means of Poincaré mapping, parameter phase portraits, equilibrium points, bifurcations and calculated Lyapunov exponents. Moreover, using RC-opamp and analog multiplier we describe a simple electronic circuit for hardware implementation of the 4-D Chua's system which differ from previously reported Chua's circuits. Various attractors of experimental results from this chaotic oscillator are in good agreement with theoretical analysis. In particular, based on the approximation theory of fractional-order operator, a relevant analog circuit diagram of this fractional-order modified Chua's system is designed with α = 0.9. Observation results demonstrate that chaos exists indeed in this fractional-order modified Chua's system with an order as low as 3.6. This fractional-order oscillation circuit, for the first time in the literature, realizes high-dimensional Chua's chaotic system.

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.

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