the corresponding Lyapunov exponent $\lambda$, and the last one, Figs. \[fig:mf-J\] (c) and Fig. \[fig:mf-W\] (c), the entropy $\eta$. The dotted vertical lines are drawn at some of the values of $J$ or $W$ where $\lambda$ passes from a negative to a positive value or vice versa. These values coincide to jumps of $\eta$ from values smaller to $1/2$ to larger ones or vice versa and mark the appearance of chaos or periodic windows in the bifurcation diagrams.

Therefore, $\eta$ can be used as a measure of chaos. To stress this, we show in Figs. \[fig:mfpdWJ\] the mean-field phase diagrams of $\lambda$ (top) and $\eta$ (bottom). These diagrams are similar. The horizontal lines at $W=15$ correspond to Figs. \[fig:mf-J\] (b) and (c) and the vertical ones at $J=-10$ to Figs. \[fig:mf-W\] (b) and (c) respectively. We find a similar behavior of the mean-field map as $K$ varies with fixed $J$ and $W$. The three quantities $J$, $W$ and $K$ are related by scaling relations, as shown in the Appendix.

![\[fig:mfpdWJ\] (Color online) (a) Mean-field phase diagram of the Lyapunov exponent $\lambda$ showing the values of $(-J,W)$ where $\lambda>0$. (b) The mean-field phase diagram of the entropy showing the values of 