scholarly journals Integer and Fractional GeneralT-System and Its Application to Control Chaos and Synchronization

2015 ◽  
Vol 2015 ◽  
pp. 1-14
Author(s):  
Mihaela Neamţu ◽  
Anamaria Liţoiu ◽  
Petru C. Strain
Keyword(s):  
Numerical Simulations ◽  
Fractional Order ◽  
Electronic Circuit ◽  
Three Dimensional ◽  
Hopf Bifurcations ◽  
Secure Communications ◽  
Bifurcation Parameter ◽  
Potential Applications ◽  
Chaos Attractor ◽  
Control And Synchronization

We propose a three-dimensional autonomous nonlinear system, called the generalTsystem, which has potential applications in secure communications and the electronic circuit. For the generalTsystem with delayed feedback, regarding the delay as bifurcation parameter, we investigate the effect of the time delay on its dynamics. We determine conditions for the existence of the Hopf bifurcations and analyze their direction and stability. Also, the fractional order generalT-system is proposed and analyzed. We provide some numerical simulations, where the chaos attractor and the dynamics of the Lyapunov coefficients are taken into consideration. The effectiveness of the chaotic control and synchronization on schemes for the new fractional order chaotic system are verified by numerical simulations.

scholarly journals Analysis of two- and three-dimensional fractional-order Hindmarsh-Rose type neuronal models

2017 ◽  
Vol 20 (3) ◽  
Author(s):  
Eva Kaslik
Keyword(s):  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Three Dimensional ◽  
Hopf Bifurcations ◽  
Bifurcation Parameter ◽  
Fast System ◽  
Stability Properties ◽  
Theoretical Results

AbstractA theoretical analysis of two- and three-dimensional fractional-order Hindmarsh-Rose neuronal models is presented, focusing on stability properties and occurrence of Hopf bifurcations, with respect to the fractional order of the system chosen as bifurcation parameter. With the aim of exemplifying and validating the theoretical results, numerical simulations are also undertaken, which reveal rich bursting behavior in the three-dimensional fractional-order slow-fast system.

HYPERCHAOS IN THE FRACTIONAL-ORDER NONAUTONOMOUS CHEN'S SYSTEM AND ITS SYNCHRONIZATION

2005 ◽  
Vol 16 (05) ◽  
pp. 815-826 ◽  
Author(s):  
HONGBIN ZHANG ◽  
CHUNGUANG LI ◽  
GUANRONG CHEN ◽  
XING GAO
Keyword(s):  
Numerical Simulations ◽  
Fractional Order ◽  
Chaotic System ◽  
Three Dimensional ◽  
Synchronization Problem ◽  
Driving Signal

Recently, a new hyperchaos generator, obtained by controlling a three-dimensional autonomous chaotic system — Chen's system — with a periodic driving signal, has been found. In this letter, we formulate and study the hyperchaotic behaviors in the corresponding fractional-order hyperchaotic Chen's system. Through numerical simulations, we found that hyperchaos exists in the fractional-order hyperchaotic Chen's system with order less than 4. The lowest order we found to have hyperchaos in this system is 3.4. Finally, we study the synchronization problem of two fractional-order hyperchaotic Chen's systems.

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.

scholarly journals Bifurcation of a Cohen-Grossberg Neural Network with Discrete Delays

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Qiming Liu ◽  
Wang Zheng
Keyword(s):  
Neural Network ◽  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Sufficient Conditions ◽  
Hopf Bifurcations ◽  
Bifurcation Parameter ◽  
Explicit Formulas ◽  
Global Hopf Bifurcation ◽  
Discrete Delays ◽  
Bifurcation And Stability

A simple Cohen-Grossberg neural network with discrete delays is investigated in this paper. The existence of local Hopf bifurcations is first considered by choosing the appropriate bifurcation parameter, and then explicit formulas are given to determine the direction of Hopf bifurcation and stability of the periodic solutions. Moreover, a set of sufficient conditions are given to guarantee the global Hopf bifurcation. Numerical simulations are given to illustrate the obtained results.

Emulating “Chaos + Chaos = Order” in Chen’s Circuit of Fractional Order by Parameter Switching

2016 ◽  
Vol 26 (06) ◽  
pp. 1650096 ◽  
Author(s):  
Wallace K. S. Tang ◽  
Marius-F. Danca
Keyword(s):  
Fractional Order ◽  
Electronic Circuit ◽  
Chaotic Circuit ◽  
Chen System ◽  
Potential Applications ◽  
Simulation And Experiment ◽  
Averaged Model ◽  
First Time ◽  
Important Design ◽  
Parameter Switching

In this paper, the effect of the parameter switching (PS) algorithm in a fractional order chaotic circuit is investigated both in simulation and experiment. The Chen system of fractional order is focused and realized in an electronic circuit. By designing a switching circuit, the PS algorithm is implemented and it is the first time, the paradoxical “Chaos [Formula: see text] Chaos [Formula: see text] Order” is presented in an electronic circuit. Both the simulation and experimental results confirm that the obtained attractor under switching approximates the attractor of the time-averaged model. Some important design issues for the circuitry realization of the PS scheme are pointed out. Finally, our work confirms the practical usage of PS algorithm in potential applications such as attractor synthesis and chaos control.

Numerical simulations of self-propelled jumping upon drop coalescence on non-wetting surfaces

2014 ◽  
Vol 752 ◽  
pp. 39-65 ◽  
Author(s):  
Fangjie Liu ◽  
Giovanni Ghigliotti ◽  
James J. Feng ◽  
Chuan-Hua Chen
Keyword(s):  
Numerical Simulations ◽  
Liquid Bridge ◽  
Flat Surface ◽  
Three Dimensional ◽  
Superhydrophobic Surfaces ◽  
Ohnesorge Number ◽  
Drop Coalescence ◽  
Out Of Plane ◽  
Potential Applications ◽  
Cutoff Radius

AbstractCoalescing drops spontaneously jump out of plane on a variety of biological and synthetic superhydrophobic surfaces, with potential applications ranging from self-cleaning materials to self-sustained condensers. To investigate the mechanism of self-propelled jumping, we report three-dimensional phase-field simulations of two identical spherical drops coalescing on a flat surface with a contact angle of 180°. The numerical simulations capture the spontaneous jumping process, which follows the capillary–inertial scaling. The out-of-plane directionality is shown to result from the counter-action of the substrate to the impingement of the liquid bridge between the coalescing drops. A viscous cutoff to the capillary–inertial velocity scaling is identified when the Ohnesorge number of the initial drops is around 0.1, but the corresponding viscous cutoff radius is too small to be tested experimentally. Compared to experiments on both superhydrophobic and Leidenfrost surfaces, our simulations accurately predict the nearly constant jumping velocity of around 0.2 when scaled by the capillary–inertial velocity. By comparing the simulated drop coalescence processes with and without the substrate, we attribute this low non-dimensional velocity to the substrate intercepting only a small fraction of the expanding liquid bridge.

scholarly journals Generalized Attracting Horseshoe in the Rössler Attractor

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 30
Author(s):  
Karthik Murthy ◽  
Ian Jordan ◽  
Parth Sojitra ◽  
Aminur Rahman ◽  
Denis Blackmore
Keyword(s):  
Differential Equations ◽  
Strange Attractor ◽  
Electronic Circuit ◽  
Three Dimensional ◽  
Dimensional System ◽  
Numerical Techniques ◽  
Circuit Realization ◽  
Trapping Region ◽  
Potential Applications ◽  
Three Dimensional System

We show that there is a mildly nonlinear three-dimensional system of ordinary differential equations—realizable by a rather simple electronic circuit—capable of producing a generalized attracting horseshoe map. A system specifically designed to have a Poincaré section yielding the desired map is described, but not pursued due to its complexity, which makes the construction of a circuit realization exceedingly difficult. Instead, the generalized attracting horseshoe and its trapping region is obtained by using a carefully chosen Poincaré map of the Rössler attractor. Novel numerical techniques are employed to iterate the map of the trapping region to approximate the chaotic strange attractor contained in the generalized attracting horseshoe, and an electronic circuit is constructed to produce the map. Several potential applications of the idea of a generalized attracting horseshoe and a physical electronic circuit realization are proposed.

scholarly journals Stability and Bifurcation of Two Kinds of Three-Dimensional Fractional Lotka-Volterra Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Jinglei Tian ◽  
Yongguang Yu ◽  
Hu Wang
Keyword(s):  
Hopf Bifurcation ◽  
Asymptotic Stability ◽  
Theoretical Analysis ◽  
Fractional Order ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Critical Value ◽  
The Other ◽  
Bifurcation Parameter ◽  
Volterra Systems

Two kinds of three-dimensional fractional Lotka-Volterra systems are discussed. For one system, the asymptotic stability of the equilibria is analyzed by providing some sufficient conditions. And bifurcation property is investigated by choosing the fractional order as the bifurcation parameter for the other system. In particular, the critical value of the fractional order is identified at which the Hopf bifurcation may occur. Furthermore, the numerical results are presented to verify the theoretical analysis.

2 × 2-SCROLL ATTRACTORS GENERATED IN A THREE-DIMENSIONAL SMOOTH AUTONOMOUS SYSTEM

2007 ◽  
Vol 17 (11) ◽  
pp. 4153-4157 ◽  
Author(s):  
WENBO LIU ◽  
WALLACE K. S. TANG ◽  
GUANRONG CHEN
Keyword(s):  
Numerical Simulations ◽  
Continuous Time ◽  
Autonomous System ◽  
Electronic Circuit ◽  
Three Dimensional ◽  
Chaotic Attractors ◽  
First Time

In this letter, a three-dimensional continuous-time smooth autonomous system with quadratic nonlinear terms is proposed for generating multiscroll chaotic attractors. Observation of 2 × 2-scroll attractors generated from this kind of system is reported for the first time. The result is confirmed by both numerical simulations and electronic circuit experiments.

scholarly journals Analysis and electronic circuit implementation of an integer-and fractional-order Shimizu-Morioka system

2021 ◽  
Vol 67 (6 Nov-Dec) ◽  
Author(s):  
François Kapche Tagne ◽  
Guillaume Honoré KOM ◽  
Marceline Motchongom Tingue ◽  
Pierre Kisito Talla ◽  
V. Kamdoum Tamba
Keyword(s):  
Numerical Simulations ◽  
Fractional Order ◽  
Close Agreement ◽  
Electronic Circuit ◽  
Dynamical Behavior ◽  
Chaotic Attractors ◽  
Transient Chaos ◽  
Circuit Implementation ◽  
Bifurcation Structures ◽  
Electronic Circuit Implementation

The dynamics of an integer-order and fractional-order Lorenz like system called Shimizu-Morioka system is investigated in this paper. It is shown thatinteger-order Shimizu-Morioka system displays bistable chaotic attractors, monostable chaotic attractors and coexistence between bistable and monostable chaotic attractors. For suitable choose of parameters, the fractional-order Shimizu-Morioka system exhibits bistable chaotic attractors, monostable chaotic attractors, metastable chaos (i.e. transient chaos) and spiking oscillations. The bifurcation structures reveal that the fractional-order derivative affects considerably the dynamics of Shimizu-Morioka system. The chain fractance circuit is used to designand implement the integer- and fractional-order Shimizu-Morioka system in Pspice. A close agreement is observed between PSpice based circuit simulations and numerical simulations analysis. The results obtained in this work were not reported previously in the interger as well as in fractional-order Shimizu-Morioka system and thus represent an important contribution which may help us in better understanding of the dynamical behavior of this class of systems.

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