Bogdanov–Takens Bifurcation in a Neutral Delayed Hopfield Neural Network with Bidirectional Connection

2020 ◽  
Vol 13 (06) ◽  
pp. 2050049
Author(s):  
Houssem Achouri ◽  
Chaouki Aouiti ◽  
Bassem Ben Hamed
Keyword(s):  
Neural Network ◽  
Normal Form ◽  
Characteristic Equation ◽  
Center Manifold ◽  
Hopfield Neural Network ◽  
Second Step ◽  
Double Zero ◽  
The Third ◽  
Dynamical Behaviors ◽  
Linearized System

In this paper, a neutral Hopfield neural network with bidirectional connection is considered. In the first step, by choosing the connection weights as parameters bifurcation, the critical point at which a zero root of multiplicity two occurs in the characteristic equation associated with the linearized system. In the second step, we studied the zeros of a third degree exponential polynomial in order to make sure that except the double zero root, all the other roots of the characteristic equation have real parts that are negative. Moreover, we find the critical values to guarantee the existence of the Bogdanov–Takens bifurcation. In the third step, the normal form is obtained and its dynamical behaviors are studied after the use of the reduction on the center manifold and the theory of the normal form. Furthermore, for the demonstration of our results, we have given a numerical example.

CODIMENSION ONE BIFURCATION OF EQUIVARIANT NEURAL NETWORK MODEL WITH DELAY

2010 ◽  
Vol 20 (04) ◽  
pp. 1255-1259
Author(s):  
CHUNRUI ZHANG ◽  
BAODONG ZHENG
Keyword(s):  
Neural Network ◽  
Normal Form ◽  
Network Model ◽  
Neural Network Model ◽  
Center Manifold ◽  
Double Zero ◽  
Codimension One ◽  
Bam Neural Network ◽  
Manifold Theory ◽  
Form Approach

In this paper, we consider double zero singularity of a symmetric BAM neural network model with a time delay. Based on the normal form approach and the center manifold theory, we obtain the normal form on the centre manifold with double zero singularity. Some numerical simulations support our analysis results.

scholarly journals Nonlinear Dynamics and Chaos in Fractional-Order Hopfield Neural Networks with Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Xia Huang ◽  
Zhen Wang ◽  
Yuxia Li
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Fractional Order ◽  
Hopfield Neural Network ◽  
Hopfield Neural Networks ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Chaotic Motions ◽  
Dynamical Behaviors ◽  
Nonlinear Dynamics And Chaos

A fractional-order two-neuron Hopfield neural network with delay is proposed based on the classic well-known Hopfield neural networks, and further, the complex dynamical behaviors of such a network are investigated. A great variety of interesting dynamical phenomena, including single-periodic, multiple-periodic, and chaotic motions, are found to exist. The existence of chaotic attractors is verified by the bifurcation diagram and phase portraits as well.

PRACTICAL COMPUTATION OF NORMAL FORMS ON CENTER MANIFOLDS AT DEGENERATE BOGDANOV–TAKENS BIFURCATIONS

2005 ◽  
Vol 15 (11) ◽  
pp. 3535-3546 ◽  
Author(s):  
YU. A. KUZNETSOV
Keyword(s):  
Normal Form ◽  
Numerical Evaluation ◽  
Center Manifold ◽  
Normal Forms ◽  
Center Manifolds ◽  
Linear Transformations ◽  
Double Zero ◽  
Practical Computation ◽  
Generalized Eigenvectors

Simple computational formulas are derived for the two-, three-, and four-order coefficients of the smooth normal form on the center manifold at the Bogdanov–Takens (nonsemisimple double-zero) bifurcation for n-dimensional systems with arbitrary n ≥ 2. These formulas are equally suitable for both symbolic and numerical evaluation and allow one to classify all codim 3 Bogdanov–Takens bifurcations in generic multidimensional ODEs. They are also applicable to systems with symmetries. We perform no preliminary linear transformations but use only critical (generalized) eigenvectors of the linearization matrix and its transpose. The derivation combines the approximation of the center manifold with the normalization on it. Three known models are used as test examples to demonstrate advantages of the method.

Dynamical Effects of Neuron Activation Gradient on Hopfield Neural Network: Numerical Analyses and Hardware Experiments

2019 ◽  
Vol 29 (04) ◽  
pp. 1930010 ◽  
Author(s):  
Bocheng Bao ◽  
Chengjie Chen ◽  
Han Bao ◽  
Xi Zhang ◽  
Quan Xu ◽  
...  
Keyword(s):  
Neural Network ◽  
Control Parameter ◽  
Period Doubling ◽  
Response Speed ◽  
Hopfield Neural Network ◽  
Activation Function ◽  
Numerical Analyses ◽  
Dynamical Behaviors ◽  
Neuron Activation ◽  
Crisis Scenarios

Hyperbolic tangent function, a bounded monotone differentiable function, is usually taken as a neuron activation function, whose activation gradient, i.e. gain scaling parameter, can reflect the response speed in the neuronal electrical activities. However, the previously published literatures have not yet paid attention to the dynamical effects of the neuron activation gradient on Hopfield neural network (HNN). Taking the neuron activation gradient as an adjustable control parameter, dynamical behaviors with the variation of the control parameter are investigated through stability analyses of the equilibrium states, numerical analyses of the mathematical model, and experimental measurements on a hardware level. The results demonstrate that complex dynamical behaviors associated with the neuron activation gradient emerge in the HNN model, including coexisting limit cycle oscillations, coexisting chaotic spiral attractors, chaotic double scrolls, forward and reverse period-doubling cascades, and crisis scenarios, which are effectively confirmed by neuron activation gradient-dependent local attraction basins and parameter-space plots as well. Additionally, the experimentally measured results have nice consistency to numerical simulations.

Multiple Time Scale Analysis for Bifurcation From a Double-Zero Eigenvalue

1999 ◽  
Author(s):  
Angelo Luongo ◽  
Achille Paolone ◽  
Angelo Di Egidio
Keyword(s):  
Normal Form ◽  
Center Manifold ◽  
Jacobian Matrix ◽  
Multiple Scale ◽  
Perturbation Parameter ◽  
Multiple Time ◽  
Double Zero ◽  
Zero Eigenvalue ◽  
Unknown Amplitude ◽  
Normal Form Method

Abstract The multiple scale method is applied to analyze bifurcations from a double zero eigenvalue of general multiparameter dynamical systems. Due to the coalescence of the eigenvalues, the Jacobian matrix at the bifurcation is nilpotent. This entails using time scales with fractional powers of the perturbation parameter. The reconstitution method is employed lo obtain a second-order o.d.e. in the unique unknown amplitude. It coincides with Bogdanova-Arnold’s normal form for the bifurcation equation. Referring to an example, the present approach and the classical center manifold plus normal form method are compared. Finally, the mechanical behavior of a non-conservative two d.o.f. system is studied.

Bifurcation Analysis of a Bounded Rational Duopoly Game with Consumer Surplus

2021 ◽  
Vol 31 (07) ◽  
pp. 2150097
Author(s):  
Wei Zhou ◽  
Yinxia Cao ◽  
Amr Elsonbaty ◽  
A. A. Elsadany ◽  
Tong Chu
Keyword(s):  
Normal Form ◽  
Theoretical Analysis ◽  
Center Manifold ◽  
Profit Maximization ◽  
Consumer Surplus ◽  
Normal Form Theory ◽  
Nonlinear Dynamical ◽  
Center Manifold Theorem ◽  
Dynamical Behaviors ◽  
Form Theory

The nonlinear dynamical behaviors of economic models have been extensively examined and still represented a great challenge for economists in recent and future years. A proposed boundedly rational game incorporating consumer surplus is introduced. This paper aims at studying stability and bifurcation types of the presented model. The flip and Neimark–Sacker bifurcations are analyzed via applying the normal form theory and the center manifold theorem. This study helps determine an appropriate choice of decision parameters which have significant influences on the behavior of the game. The duopoly game that is formed by considering bounded rationality and consumer surplus is more realistic than the ordinary duopoly game which only has profit maximization. And then, some numerical simulations are provided to verify the theoretical analysis. Finally, we compare the dynamical behaviors of the built model with that of Bischi–Naimzada model so as to better understand the performance of the duopoly game with consumer surplus.

scholarly journals The Dynamical Behaviors for a Class of Immunogenic Tumor Model with Delay

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Ping Bi ◽  
Zijian Liu ◽  
Mutei Damaris Muthoni ◽  
Jianhua Pang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Tumor Model ◽  
Numerical Examples ◽  
Dynamical Behaviors ◽  
Existence And Stability ◽  
Discrete Delays

This paper aims at studying the model proposed by Kuznetsov and Taylor in 1994. Inspired by Mayer et al., time delay is introduced in the general model. The dynamic behaviors of this model are studied, which include the existence and stability of the equilibria and Hopf bifurcation of the model with discrete delays. The properties of the bifurcated periodic solutions are studied by using the normal form on the center manifold. Numerical examples and simulations are given to illustrate the bifurcation analysis and the obtained results.

scholarly journals Three-stage question answering system with sentence ranking

10.29007/bj7w ◽  
2019 ◽  
Author(s):  
Daria Soboleva ◽  
Konstantin Vorontsov
Keyword(s):  
Neural Network ◽  
High Speed ◽  
Question Answering ◽  
Feature Space ◽  
Second Step ◽  
Question Answering System ◽  
The Third ◽  
Construction Techniques ◽  
Space Construction ◽  
Speed Up

We explore a recently proposed question answering system. We developed a high speed modification based on dividing the question answering system into three consecutive stages. The first step is to find the candidate documents that most likely contain the answer to the question. The second step is to rank sentences by the probability of having a correct answer to the question. The third step is to find the exact phrase that answers the question. At the third step we used a recently proposed recurrent bidirectional neural network predicting the beginning and the end of a response. In this paper we showed that the proposed question answering system allows to speed up its work without significant losses in the quality. For each step we also explored the feature space construction techniques allowing to improve the final quality.

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