to period 2, $\eta=1/8$, and period 4, $\eta=1/4$, orbits. For every value of $J$, two initial values of the magnetization were used, $m_0=-0.3$ and $m_0=0.3$.](mf-m-lambda-eta-PI00-o2-k20-W15p00){width="0.9\columnwidth"}

![\[fig:mf-W\] (Color online.) (a) Bifurcation diagram of the mean-field map of the magnetization $m$, Eq. , as a function of the coupling constant $W$ with $J=-10$. In (b) The corresponding Lyapunov exponent $\lambda$ and in (c) the entropy $\eta$. The vertical dotted lines are drawn at the estimated values of $W$ for which $\lambda=0$. The horizontal dotted lines in (c) correspond to period 2, $\eta=1/8$, and period 4, $\eta=1/4$, orbits. For every value of $W$, two initial values of the magnetization were used, $m_0=-0.3$ and $m_0=0.3$.](mf-lambda-eta-PI00-o13--14-k20-J-10){width="0.9\columnwidth"}

In Fig. \[fig:mf-J\] (a) we show the bifurcation diagram of the magnetization $m$, Eq. , as a function of $J$ with $W$ and $K$ fixed. The diagram exhibits a period doubling cascade towards chaos with periodic windows and pitchfork bifurcations.

The bifurcation diagram of $m$ as a function of $W$ with $J$ and $K$ fixed is shown in Fig. \[fig:mf-W\] (a). In this case, there is an inverse period doubling cascade to chaos with pitchfork bifurcations.

The next row of figures, Figs. \[fig:mf-J\] (b) and Fig. \[fig:mf-W\] (b), show 