Circle maps with gaps: Understanding the dynamics of the two-process model for sleep–wake regulation

For more than thirty years the `two process model' has played a central role in the under- standing of sleep/wake regulation. This ostensibly simple model is an interesting example of a nonsmooth dynamical system whose rich dynamical structure has been relatively un- explored. The two process model can be framed as a one-dimensional map of the circle which, for some parameter regimes, has gaps. We show how border collision bifurcations that arise naturally in maps with gaps extend and supplement the Arnold tongue saddle- node bifurcation set that is a feature of continuous circle maps. The novel picture that results shows how the periodic solutions that are created by saddle-node bifurcations in continuous maps transition to periodic solutions created by period-adding bifurcations as seen in maps with gaps.