Dynamics of a class of neutral three neurons network with delay
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RESEARCH
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Dynamics of a class of neutral three neurons network with delay Ming Liu, Chunrui Zhang* and Xiaofeng Xu * Correspondence: [email protected] Department of mathematics, Northeast Forestry University, Harbin, 150040, P.R. China
Abstract In this paper, a class of neutral neural networks with delays is investigated. The linear stability of the model is studied. It is found that a Hopf bifurcation also occurs when some delays pass through a sequence of critical values. The direction of the Hopf bifurcations and the stability of bifurcating periodic solutions are determined by using the normal form method and center manifold theory. The existence of a permanent oscillation is established using Chafee’s criterion. Numerical simulations are performed to support the analytical results. Keywords: neutral neural network; stability; Hopf bifurcation; permanent oscillation
1 Introduction Since s, the theories and applications of neural networks with delays have been greatly developed. It is well known that many important mathematical models from physics, biology, etc. can be written in neurons network models. In , Li and Yuan considered a Hopfield-type network of three identical neurons coupled in any possible way in []: x˙ (t) = –x (t) + af x (t – τs ) + a bg x (t – τn ) + a bg x (t – τn ) , x˙ (t) = –x (t) + a bg x (t – τn ) + af x (t – τs ) + a bg x (t – τn ) , x˙ (t) = –x (t) + a bg x (t – τn ) + a bg x (t – τn ) + af x (t – τs ) .
(.)
Due to the finite speed of the switching and transmission of signals, neutral behavior does exist in the neural network with delays and should be incorporated. For this reason, we improve the original model in which the neutral behavior was added and obtain the following forms []: x˙ (t) = –x (t) + af x (t – τs ) + a bg x (t – τn ) + a bg x (t – τn ) + a f x˙ (t – τs ) + b b g x˙ (t – τn ) + b b g x˙ (t – τn ) , x˙ (t) = –x (t) + a bg x (t – τn ) + af x (t – τs ) + a bg x (t – τn ) + b b g x˙ (t – τn ) + a f x˙ (t – τs ) + b b g x˙ (t – τn ) , x˙ (t) = –x (t) + a bg x (t – τn ) + a bg x (t – τn ) + af x (t – τs ) + b b g x˙ (t – τn ) + b b g x˙ (t – τn ) + a f x˙ (t – τs ) ,
(.)
©2013 Liu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Liu et al. Advances in Difference Equations 2013, 2013:338 http://www.advancesindifferenceequations.com/content/2013/1/338
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where aij , bij (i = j, i, j = , , ) have the values or , depending whether the cells from j to i are connected or not; a, b, a , b ∈ R denote the strength in self-connection and neighboring-connection, respectively; τs
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