Multiple Cusp Bifurcation

Abstract

In Section 5.3.3 we showed that any weakly connected neural network of the form

$$ \dot X_i = F_i \left( {X_i ,\lambda } \right) + \varepsilon G_i \left( {X,\lambda ,\rho ,\varepsilon } \right) $$

at a multiple cusp bifurcation is governed by the canonical model

$$ x'_i = r_i + b_i x_i + \sigma _i x_i^3 + \sum\limits_{j = 1}^n {c_{ij} } x_{ij} ,{\text{ i = 1,}} \ldots {\text{,n,}} $$

((11.1))

where ′ = d/dτ, τ is slow time, x _i , r _i , b _i c _ij are real variables, and σ _i = ±1. In this chapter we study some neurocomputational properties of this canonical model. In particular, we use Hirsch’s theorem to prove that the canonical model can work as a globally asymptotically stable neural network (GAS-type NN) for certain choices of the parameters b₁,…, b _n . We use Cohen and Grossberg’s convergence theorem to show that the canonical model can also operate as a multiple attractor neural network (MA-type NN).