The one-dimensional map for the two-process model.

<p>(a) A single trajectory of the two-process model showing successive times of sleep onset. (b) Trajectories of the two-process model for different initial sleep onset times. Each different sleep onset time results in a different sequence, , but each sequence rapidly converges to the same sleep onset time, of modulo 1 day. (c) A zoom of (b), showing only the trajectories for and (d) First return map for the two-process model. The black line shows the return map, in other words for any value of sleep onset time on day , it shows the onset time of sleep on day , . The grey diagonal line is the line along which . One typical trajectory is plotted for showing the rapid convergence to the periodic cycle where modulo 1 day, the point at which the return map and the diagonal line intersect. The discontinuity is a result of the fact that neighbouring values of exist that lead to very different values for , as shown in (c). Parameter values for the two-process model are based on the PR model for the human sleep-wake cycle and can be found in (15).