Chaotic behavior in a learning model (Q1049457)

The authors studies a learning model defined by the system of difference equations \[ \begin{aligned} a^{k+1} &= a^k + [W^k a^k - \gamma a^k + e, \\ W^{k+1} &= W^k + \beta [e - W^k a^k] \odot (a^k)^T, \end{aligned} \] where the product \(\odot\) is defined by \[ (x \odot y^{T})_{ij} = \begin{cases} x_i y_j & \text{if } i \neq j,\\ 0 & \text{if } i = j. \end{cases} \] Some chaotic behaviour is observed.