CHAOS IN A THREE-DIMENSIONAL GENERAL MODEL OF NEURAL NETWORK

2002 ◽  
Vol 12 (10) ◽  
pp. 2271-2281 ◽  
Author(s):  
A. DAS ◽  
PRITHA DAS ◽  
A. B. ROY
Keyword(s):  
Neural Network ◽  
Differential Equations ◽  
General Model ◽  
Three Dimensional ◽  
Stationary Regime ◽  
Decay Rates ◽  
Phase Portraits ◽  
Study Phase ◽  
Bifurcation Diagrams ◽  
Parameter Values

The dynamics of a network of three neurons with all possible connections is studied here. The equations of control are given by three differential equations with nonlinear, positive and bounded sigmoidal response function of the neurons. The system passes from stable to periodic and then to chaotic regimes and returns to stationary regime with change in parameter values of synaptic weights and decay rates. We have developed programs and used Locbif package to study phase portraits, bifurcation diagrams which confirm the result. Lyapunov Exponents have been calculated to confirm chaos.

scholarly journals Dynamics Analysis of a New Fractional-Order Hopfield Neural Network with Delay and Its Generalized Projective Synchronization

Entropy ◽  
2018 ◽  
Vol 21 (1) ◽  
pp. 1 ◽  
Author(s):  
Han-Ping Hu ◽  
Jia-Kun Wang ◽  
Fei-Long Xie
Keyword(s):  
Neural Network ◽  
Fractional Order ◽  
Complex Dynamics ◽  
Three Dimensional ◽  
Hopfield Neural Network ◽  
Projective Synchronization ◽  
Phase Portraits ◽  
Largest Lyapunov Exponent ◽  
Bifurcation Diagrams ◽  
Intermittent Chaos

In this paper, a new three-dimensional fractional-order Hopfield-type neural network with delay is proposed. The system has a unique equilibrium point at the origin, which is a saddle point with index two, hence unstable. Intermittent chaos is found in this system. The complex dynamics are analyzed both theoretically and numerically, including intermittent chaos, periodicity, and stability. Those phenomena are confirmed by phase portraits, bifurcation diagrams, and the Largest Lyapunov exponent. Furthermore, a synchronization method based on the state observer is proposed to synchronize a class of time-delayed fractional-order Hopfield-type neural networks.

Nonlinear Dynamics and Application of the Four Dimensional Autonomous Hyper-Chaotic System

2014 ◽  
Author(s):  
Ge Kai ◽  
Wei Zhang
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Simulation ◽  
Differential Equations ◽  
Chaotic System ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Phase Portraits ◽  
State Variables ◽  
Chaotic Motions ◽  
Finance System

In this paper, we establish a dynamic model of the hyper-chaotic finance system which is composed of four sub-blocks: production, money, stock and labor force. We use four first-order differential equations to describe the time variations of four state variables which are the interest rate, the investment demand, the price exponent and the average profit margin. The hyper-chaotic finance system has simplified the system of four dimensional autonomous differential equations. According to four dimensional differential equations, numerical simulations are carried out to find the nonlinear dynamics characteristic of the system. From numerical simulation, we obtain the three dimensional phase portraits that show the nonlinear response of the hyper-chaotic finance system. From the results of numerical simulation, it is found that there exist periodic motions and chaotic motions under specific conditions. In addition, it is observed that the parameter of the saving has significant influence on the nonlinear dynamical behavior of the four dimensional autonomous hyper-chaotic system.

scholarly journals The Yamada Model for a Self-Pulsing Laser: Bifurcation Structure for Nonidentical Decay Times of Gain and Absorber

2020 ◽  
Vol 30 (14) ◽  
pp. 2030039
Author(s):  
Robert Otupiri ◽  
Bernd Krauskopf ◽  
Neil G. R. Broderick
Keyword(s):  
Three Dimensional ◽  
Equation Model ◽  
Decay Rates ◽  
Phase Portraits ◽  
Stable Phase ◽  
Bifurcation Structure ◽  
Decay Times ◽  
Comprehensive Picture ◽  
Two Parameter ◽  
Parameter Bifurcation

We consider self-pulsing in lasers with a gain section and an absorber section via a mechanism known as [Formula: see text]-switching, as described mathematically by the Yamada ordinary differential equation model for the gain, the absorber and the laser intensity. More specifically, we are interested in the case that gain and absorber decay on different time-scales. We present an overall bifurcation structure by showing how the two-parameter bifurcation diagram in the plane of pump strength versus decay rate of the gain changes with the ratio between the two decay rates. In total, there are ten cases BI to BX of qualitatively different two-parameter bifurcation diagrams, which we present with an explanation of the transitions between them. Moroever, we show for each of the associated eleven cases of structurally stable phase portraits (in open regions of the parameter space) a three-dimensional representation of the organization of phase space by the two-dimensional manifolds of saddle equilibria and saddle periodic orbits. The overall bifurcation structure provides a comprehensive picture of the observable dynamics, including multistability and excitability, which we expect to be of relevance for experimental work on [Formula: see text]-switching lasers with different kinds of saturable absorbers.

scholarly journals HYPERCHAOTIC DYNAMICS OF A NEW FRACTIONAL DISCRETE-TIME SYSTEM

Fractals ◽  
2021 ◽  
pp. 2140034
Author(s):  
AMINA-AICHA KHENNAOUI ◽  
ADEL OUANNAS ◽  
SHAHER MOMANI ◽  
ZOHIR DIBI ◽  
GIUSEPPE GRASSI ◽  
...  
Keyword(s):  
Discrete Time ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Discrete Time System ◽  
Time System ◽  
Discrete Time Systems ◽  
Simulation Results ◽  
Time Systems ◽  
First Time

In recent years, some efforts have been devoted to nonlinear dynamics of fractional discrete-time systems. A number of papers have so far discussed results related to the presence of chaos in fractional maps. However, less results have been published to date regarding the presence of hyperchaos in fractional discrete-time systems. This paper aims to bridge the gap by introducing a new three-dimensional fractional map that shows, for the first time, complex hyperchaotic behaviors. A detailed analysis of the map dynamics is conducted via computation of Lyapunov exponents, bifurcation diagrams, phase portraits, approximated entropy and [Formula: see text] complexity. Simulation results confirm the effectiveness of the approach illustrated herein.

scholarly journals Generating Multidirectional Variable Hidden Attractors via Newly Commensurate and Incommensurate Non-Equilibrium Fractional-Order Chaotic Systems

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 261
Author(s):  
Nadjette Debbouche ◽  
Shaher Momani ◽  
Adel Ouannas ◽  
’Mohd Taib’ Shatnawi ◽  
Giuseppe Grassi ◽  
...  
Keyword(s):  
Fractional Order ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Phase Portraits ◽  
Hidden Attractors ◽  
Signum Function ◽  
Parameter Values ◽  
Fractional Orders ◽  
Non Equilibrium

This article investigates a non-equilibrium chaotic system in view of commensurate and incommensurate fractional orders and with only one signum function. By varying some values of the fractional-order derivative together with some parameter values of the proposed system, different dynamical behaviors of the system are explored and discussed via several numerical simulations. This system displays complex hidden dynamics such as inversion property, chaotic bursting oscillation, multistabilty, and coexisting attractors. Besides, by means of adapting certain controlled constants, it is shown that this system possesses a three-variable offset boosting system. In conformity with the performed simulations, it also turns out that the resultant hidden attractors can be distributively ordered in a grid of three dimensions, a lattice of two dimensions, a line of one dimension, and even arbitrariness in the phase space. Through considering the Caputo fractional-order operator in all performed simulations, phase portraits in two- and three-dimensional projections, Lyapunov exponents, and the bifurcation diagrams are numerically reported in this work as beneficial exit results.

Bifurcation Analysis of a One-Prey and Two-Predators Model with Additional Food and Harvesting Subject to Toxicity

2021 ◽  
Vol 31 (06) ◽  
pp. 2150089
Author(s):  
Biruk Tafesse Mulugeta ◽  
Liping Yu ◽  
Jingli Ren
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Additional Food ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Generalized Hopf Bifurcation ◽  
Form Theory

In this paper, a three-dimensional one-prey and two-predators model, with additional food and harvesting in the presence of toxicity is proposed. Additional food is being provided to one predator. The dynamics and bifurcations of the system are investigated using center manifold theorem, normal form theory and Sotomayor’s theorem. It is proved that the system undergoes transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, generalized Hopf bifurcation, Bogdanov–Takens bifurcation and cusp bifurcation with respect to different parameters. Bifurcation diagrams of the system with respect to toxic effect and harvesting effect are illustrated. The phase portraits and solution curves are also presented to verify the dynamic behavior. The results show that the combined effect of the factors has the power of transforming simple ecosystems into complex ecosystems.

scholarly journals Nonlinear Dynamic Analysis of Macrofiber Composites Laminated Shells

2017 ◽  
Vol 2017 ◽  
pp. 1-17 ◽  
Author(s):  
Xiangying Guo ◽  
Dameng Liu ◽  
Wei Zhang ◽  
Lin Sun ◽  
Shuping Chen
Keyword(s):  
Differential Equations ◽  
Nonlinear Dynamic ◽  
Dynamic Models ◽  
Three Dimensional ◽  
Nonlinear Dynamic Analysis ◽  
Dynamical Analysis ◽  
Phase Portraits ◽  
Chaotic Motions ◽  
Laminated Shells ◽  
Laminated Shell

This work presents the nonlinear dynamical analysis of a multilayer d31 piezoelectric macrofiber composite (MFC) laminated shell. The effects of transverse excitations and piezoelectric properties on the dynamic stability of the structure are studied. Firstly, the nonlinear dynamic models of the MFC laminated shell are established. Based on known selected geometrical and material properties of its constituents, the electric field of MFC is presented. The vibration mode-shape functions are obtained according to the boundary conditions, and then the Galerkin method is employed to transform partial differential equations into two nonlinear ordinary differential equations. Next, the effects of the transverse excitations on the nonlinear vibration of MFC laminated shells are analyzed in numerical simulation and moderating effects of piezoelectric coefficients on the stability of the system are also presented here. Bifurcation diagram, two-dimensional and three-dimensional phase portraits, waveforms phases, and Poincare diagrams are shown to find different kinds of periodic and chaotic motions of MFC shells. The results indicate that piezoelectric parameters have strong effects on the vibration control of the MFC laminated shell.

BEHAVIORAL DYNAMICS OF TWO INTERACTING HAWK–DOVE POPULATIONS

2001 ◽  
Vol 11 (04) ◽  
pp. 645-661 ◽  
Author(s):  
PIERRE AUGER ◽  
RAFAEL BRAVO DE LA PARRA ◽  
EVA SÁNCHEZ
Keyword(s):  
General Model ◽  
The Other ◽  
Phase Portraits ◽  
Behavioral Dynamics ◽  
Interacting Populations ◽  
Different Populations ◽  
The Stability ◽  
Individual Strategies ◽  
Parameter Values ◽  
Two Populations

We present a model of two interacting populations using two individual strategies, hawk and dove. Individuals encounter each other frequently and can change tactics several times in their life. Conflicts occur between individuals belonging to the same population and to different populations. The general model is based on the replicator equations which are used to describe the variations of the hawk proportions of the two populations. According to parameter values, namely the gain-, the intra- and inter-population costs, and the relative intra-population encounter rates, we classify the different phase portraits. We show that a decrease in the intra-population cost of a population provokes an increase in the hawk proportion in this population and of the dove proportion in the other population. An increase in the inter-population cost favors hawk strategy in the population which causes more injuries and dove strategy in the other. We also study the effects of the relative densities of the two populations on the stability of equilibria. In most cases, an increase in the relative density of a population leads to a decrease in hawk proportion in this population and of dove proportion in the other.

Detecting Parameter Changes Using Experimental Nonlinear Dynamics and Chaos

1996 ◽  
Vol 118 (3) ◽  
pp. 375-383 ◽  
Author(s):  
R. S. Chancellor ◽  
R. M. Alexander ◽  
S. T. Noah
Keyword(s):  
Nonlinear Dynamics ◽  
Periodic Orbits ◽  
Piecewise Linear ◽  
Phase Portraits ◽  
Unstable Periodic Orbits ◽  
Bifurcation Diagrams ◽  
Frequency Spectra ◽  
Nonlinear Dynamics And Chaos ◽  
Chaotic Nature ◽  
Parameter Values

A method of detecting parameter changes using analytical and experimental nonlinear dynamics and chaos is applied to a piecewise-linear oscillator. Experimental data show the chaotic nature of the system through phase portraits, Poincare´ maps, frequency spectra and bifurcation diagrams. Unstable periodic orbits were extracted from each chaotic time series obtained from the system with six different parameter values. Movement of the unstable periodic orbits in phase space is used to detect parameter changes in the system.

A Memristive Hyperchaotic Multiscroll Jerk System with Controllable Scroll Numbers

2017 ◽  
Vol 27 (06) ◽  
pp. 1750091 ◽  
Author(s):  
Chunhua Wang ◽  
Hu Xia ◽  
Ling Zhou
Keyword(s):  
Three Dimensional ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Complex Dynamic ◽  
Coexisting Attractors ◽  
Circuit Element ◽  
Circuit Structure ◽  
Jerk System ◽  
New System ◽  
System Phase

A memristor is the fourth circuit element, which has wide applications in chaos generation. In this paper, a four-dimensional hyperchaotic jerk system based on a memristor is proposed, where the scroll number of the memristive jerk system is controllable. The new system is constructed by introducing one extra flux-controlled memristor into three-dimensional multiscroll jerk system. We can get different scroll attractors by varying the strength of memristor in this system without changing the circuit structure. Such a method for controlling the scroll number without changing the circuit structure is very important in designing the modern circuits and systems. The new memristive jerk system can exhibit a hyperchaotic attractor, which has more complex dynamic behavior. Furthermore, coexisting attractors are observed in the system. Phase portraits, dissipativity, equilibria, Lyapunov exponents and bifurcation diagrams are analyzed. Finally, the circuit implementation is carried out to verify the new system.

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