Abstract
A kind of 2-dimensional neural network model with delay is considered. By analyzing the distribution of the roots of the characteristic equation associated with the model, a bifurcation diagram was drawn in an appropriate parameter plane. It is found that a line is a pitchfork bifurcation curve. Further more, the stability of each fixed point and existence of Hopf bifurcation were obtained. Finally, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions were determined by using the normal form method and centre manifold theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chen Y, Wu J. Slowly oscillating periodic solutions for a delayed frustrated network of two neurons [J].J Math Anal Appl, 2001,259(1):188–208.
Wei J, Ruan S. Stability and bifurcation in a neural network model with two delays[J].Physica D, 1999,130(3/4):255–272.
Faria T. On a planar system modelling a neuron network with memory[J].J Differential Equations, 2000,168(1):129–149.
Wei J, Velarde M, Makarov V. Oscillatory phenomena and stability of periodic solutions in a simple neural network with delay[J].Nonlinear Phenomena in Complex Systems, 2002,5(4):407–417.
Wu J. Symmetric functional differential equations and neural networks with memory[J].Trans Amer Math Soc, 1998,350(12):4799–4838.
Wu J.Introduction to Neural Dynamics and Signal Transmission Delay[M]. Walter de Gruyter, Berlin, New York, 2001, 120–150.
Babcock K L, Westervelt R M. Dynamics of simple electronic neural networks[J].Physica D, 1987,28(4):305–359.
Hassard B D, Kazarinoff N D, Wan Y H.Theory and Applications of Hopf Bifurcation[M]. Cambridge University Press, Cambridge, 1981.
Author information
Authors and Affiliations
Additional information
Communicated by LIN Zong-chi, Original Member of Editorial Committee, AMM
Foundation item: the National Natural Science, Foundation of China (19831030)
Biography: WEI Jun-jie, Professor, Doctor, E-mail: weijj@hit.edu.cn
Rights and permissions
About this article
Cite this article
Jun-jie, W., Chun-rui, Z. & Xiu-ling, L. Bifurcation in two-dimensional neural network model with delay. Appl Math Mech 26, 210–217 (2005). https://doi.org/10.1007/BF02438244
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02438244