  The ability of a circadian system to entrain to the 24-hour light-dark cycle
is one of its most important properties. A new tool, called the entrainment
map, was recently introduced to study this process for a single oscillator.
Here we generalize the map to study the effects of light-dark forcing in a
hierarchical system consisting of a central circadian oscillator that drives a
peripheral circadian oscillator. We develop techniques to reduced the higher
dimensional phase space of the coupled system to derive a generalized 2-D
entrainment map. Determining the nature of various fixed points, together with
an understanding of their stable and unstable manifolds, leads to conditions
for existence and stability of periodic orbits of the circadian system. We use
the map to investigate how various properties of solutions depend on parameters
and initial conditions including the time to and direction of entrainment. We
show that the concepts of phase advance and phase delay need to be carefully
assessed when considering hierarchical systems.
