Analysis of a distributed model of leg coordination

This is the first paper that I am reading sentence by sentence, underlying every sentence that I don't understand. There are many underlines. Presumably, within the next six months, this sentence will be transparent:

"Specifically, we derive analytical return maps for each of the mechanisms, study the asymptotic form of these return maps, and characterize their layout in parameter space. In addition, we demonstrate how to calculate the asymtotic relative phase density for each mechanism"

Some papers to read: Dean 1991a,b, 1992a,b "extensive simulation studies" of 'this' model of leg coordination Cruse et al 1995 "Six different mechanisms have been described for the stick insect..." Espenschied et al. 1993 "...however, a subset of three of these mechanisms...is known to be sufficient for generating a full range of insect-like gaits."