Hopf Bifurcation Analysis for a Four-Dimensional Recurrent Neural Network with Two Delays

Hopf Bifurcation Analysis for a Four-Dimensional Recurrent Neural Network with Two Delays

A four-dimensional recurrent neural network with two delays is considered. The main result is given in terms of local stability and Hopf bifurcation. Sufficient conditions for local stability of the zero equilibrium and existence of the Hopf bifurcation with respect to both delays are obtained by an...

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Main Authors: Zizhen Zhang, Huizhong Yang
Format: Article
Language:English
Published: Hindawi Limited 2013-01-01
Series:Journal of Applied Mathematics
Online Access:http://dx.doi.org/10.1155/2013/436254
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spelling doaj-b8d0837e75db4092836a1e7ec485805d2020-11-24T23:13:40ZengHindawi LimitedJournal of Applied Mathematics1110-757X1687-00422013-01-01201310.1155/2013/436254436254Hopf Bifurcation Analysis for a Four-Dimensional Recurrent Neural Network with Two DelaysZizhen Zhang0Huizhong Yang1Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Jiangnan University, Wuxi 214122, ChinaKey Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Jiangnan University, Wuxi 214122, ChinaA four-dimensional recurrent neural network with two delays is considered. The main result is given in terms of local stability and Hopf bifurcation. Sufficient conditions for local stability of the zero equilibrium and existence of the Hopf bifurcation with respect to both delays are obtained by analyzing the distribution of the roots of the associated characteristic equation. In particular, explicit formulae for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are established by using the normal form theory and center manifold theory. Some numerical examples are also presented to verify the theoretical analysis.http://dx.doi.org/10.1155/2013/436254
collection DOAJ
language English
format Article
sources DOAJ
author Zizhen Zhang
Huizhong Yang
spellingShingle Zizhen Zhang
Huizhong Yang
Hopf Bifurcation Analysis for a Four-Dimensional Recurrent Neural Network with Two Delays
Journal of Applied Mathematics
author_facet Zizhen Zhang
Huizhong Yang
author_sort Zizhen Zhang
title Hopf Bifurcation Analysis for a Four-Dimensional Recurrent Neural Network with Two Delays
title_short Hopf Bifurcation Analysis for a Four-Dimensional Recurrent Neural Network with Two Delays
title_full Hopf Bifurcation Analysis for a Four-Dimensional Recurrent Neural Network with Two Delays
title_fullStr Hopf Bifurcation Analysis for a Four-Dimensional Recurrent Neural Network with Two Delays
title_full_unstemmed Hopf Bifurcation Analysis for a Four-Dimensional Recurrent Neural Network with Two Delays
title_sort hopf bifurcation analysis for a four-dimensional recurrent neural network with two delays
publisher Hindawi Limited
series Journal of Applied Mathematics
issn 1110-757X
1687-0042
publishDate 2013-01-01
description A four-dimensional recurrent neural network with two delays is considered. The main result is given in terms of local stability and Hopf bifurcation. Sufficient conditions for local stability of the zero equilibrium and existence of the Hopf bifurcation with respect to both delays are obtained by analyzing the distribution of the roots of the associated characteristic equation. In particular, explicit formulae for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are established by using the normal form theory and center manifold theory. Some numerical examples are also presented to verify the theoretical analysis.
url http://dx.doi.org/10.1155/2013/436254
work_keys_str_mv AT zizhenzhang hopfbifurcationanalysisforafourdimensionalrecurrentneuralnetworkwithtwodelays
AT huizhongyang hopfbifurcationanalysisforafourdimensionalrecurrentneuralnetworkwithtwodelays
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