Equilibria and Their Bifurcations in a Recurrent Neural Network Involving Iterates of a Transcendental Function

Some practical models contain so complicated mathematical expressions that it is hard to determine the number and distribution of all equilibria, not mentioning the qualitative properties and bifurcations of those equilibria. The three-node recurrent neural network system with two free weight parameters, originally introduced by Ruiz, Owens, and Townley in 1997, is such a system, for which the equation of equilibria involves transcendental function tanh(x) and its iterates. Not computing coordinates of its equilibria, in this paper, we display an effective technique to determine the number and distribution of its equilibria. Without full information about equilibria, our method enables to further study qualitative properties of those equilibria and discuss their saddle node, pitchfork, and Hopf bifurcations by approximating center manifolds.