are four basins of attraction. For small values of the initial magnetization $m_0$, the orbit is attracted to $m=-1$ and for large $m_0$ to $m=1$. The regions where this occurs are marked by the vertical dotted lines in the Figure. For other values of $m_0$, $m$ ends in one of two period-two orbits. This figure shows, in the lower part, the two basins of attraction that are symmetric in the sense that if $m_0$ belongs to one basin of attraction, $-m_0$ belongs to the other one.

In what follows we present the bifurcation diagrams of the mean-field map Eq.  by varying the coupling constants $J$ and $W$. Unless otherwise noticed, we computed the Lyapunov exponent $\lambda$ by averaging over 10,000 time steps after a transient of another $10,000$ steps. The entropy $\eta$ was computed using 256 boxes and $25,000$ time steps.

![\[fig:mf-J\] (Color online.) (a) Bifurcation diagram of the mean-field map of the magnetization $m$, Eq. , as a function of the linear coupling $J$. In (b) the corresponding Lyapunov exponent $\lambda$ and in (c) the entropy $\eta$. The vertical dotted lines are drawn at the estimated values of $J$ for which $\lambda=0$. The horizontal dotted lines in (c) correspond 