The nonresonant double Hopf bifurcation in delayed neural network

2008 ◽  
Vol 85 (6) ◽  
pp. 925-935 ◽  
Author(s):  
J. Xu ◽  
L. J. Pei
Keyword(s):  
Neural Network ◽  
Hopf Bifurcation ◽  
Double Hopf Bifurcation ◽  
Delayed Neural Network

STABILITY AND BIFURCATION ANALYSIS ON A DELAYED NEURAL NETWORK MODEL

2005 ◽  
Vol 15 (09) ◽  
pp. 2883-2893 ◽  
Author(s):  
XIULING LI ◽  
JUNJIE WEI
Keyword(s):  
Neural Network ◽  
Hopf Bifurcation ◽  
Network Model ◽  
Neural Network Model ◽  
Center Manifold ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Delayed Neural Network ◽  
The Stability

A simple delayed neural network model with four neurons is considered. Linear stability of the model is investigated by analyzing the associated characteristic equation. It is found that Hopf bifurcation occurs when the sum of four delays varies and passes a sequence of critical values. The stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. An example is given and numerical simulations are performed to illustrate the obtained results. Meanwhile, the bifurcation set is provided in the appropriate parameter plane.

Double Hopf Bifurcation and the Existence of Quasi-Periodic Invariant Tori in a Generalized Gopalsamy Delayed Neural Network Model

2019 ◽  
Vol 29 (13) ◽  
pp. 1950182
Author(s):  
Fang Wu ◽  
Xuemei Li
Keyword(s):  
Neural Network ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Network Model ◽  
Neural Network Model ◽  
Invariant Tori ◽  
Sufficient Conditions ◽  
Small Neighborhood ◽  
Double Hopf Bifurcation ◽  
Truncated Normal

In this paper, we discuss the double Hopf bifurcation and the existence of quasi-periodic invariant tori in a generalized Gopalsamy neural network model. Regarding the connection weight and the delay as bifurcation parameters of the double Hopf bifurcation, we derive the normal form up to the fifth order near the critical point by using the center manifold theorem and the normal form method, and obtain sufficient conditions on the existence of invariant 2-tori for the truncated normal form. Moreover, we investigate the effect of higher-order terms on these 2-tori by a KAM theorem. It is proved that in a sufficiently small neighborhood of the bifurcation point, the neural network model has quasi-periodic invariant 2-tori for most of the parameter set where its truncated normal form possesses invariant 2-tori. We give a numerical example to verify the conditions on all results in remarks.

On control of Hopf bifurcation in time-delayed neural network system

2005 ◽  
Vol 338 (3-5) ◽  
pp. 261-271 ◽  
Author(s):  
Shangbo Zhou ◽  
Xiaofeng Liao ◽  
Juebang Yu ◽  
Kwok-wo Wong
Keyword(s):  
Neural Network ◽  
Hopf Bifurcation ◽  
Network System ◽  
Neural Network System ◽  
Delayed Neural Network ◽  
Time Delayed Neural Network

scholarly journals Dynamical Behavior in a Four-Dimensional Neural Network Model with Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Changjin Xu ◽  
Peiluan Li
Keyword(s):  
Neural Network ◽  
Hopf Bifurcation ◽  
Network Model ◽  
Neural Network Model ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Normal Form Theory ◽  
Form Theory ◽  
Explicit Formulae ◽  
Delay Differential

A four-dimensional neural network model with delay is investigated. With the help of the theory of delay differential equation and Hopf bifurcation, the conditions of the equilibrium undergoing Hopf bifurcation are worked out by choosing the delay as parameter. Applying the normal form theory and the center manifold argument, we derive the explicit formulae for determining the properties of the bifurcating periodic solutions. Numerical simulations are performed to illustrate the analytical results.

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