The Bogdanov–Takens bifurcation analysis on a three dimensional recurrent neural network

Abstract

A class of recurrent neural networks is investigated in the vicinity of the Bogdanov–Takens bifurcation point in the parameter space when the slope of the transfer function of the neurons at the origin is not equal to one. It will be shown that two different bifurcation diagrams can be constructed. In each bifurcation diagram, there are critical values for the parameters of the network for which curves of pitchfork and Hopf bifurcation intersect each other at a point where the linear part of the system that describes the network, has a pair of simple zero eigenvalues. As curves of homoclinic and heteroclinic bifurcation emanate from the Bogdanov–Takens point, a complicated behavior is observed by the variation of weights in the recurrent neural network.