A central feature of circadian systems is their response to an external, pacemaking 24 hour light-dark (LD) drive which typically leads to entrainment of circadian oscillator. There are, however, several naturally arising situations in which a circadian system is incapable of entrainment, either due to abnormal intrinsic properties of the oscillators or due to changes in the light-dark input that the oscillators receive. In this talk, we will use entrainment maps to describe circumstances that fall outside the normal fixed phase relationship between LD forcing and oscillator such as during jet lag, shift work and non-24 hour sleep-wake disorder. The mathematical and computational methods used to study these problems revolve around finding stable limit cycle solutions of the governing equations and it is the reduction of this study to a one-dimensional framework that will be the focus of the talk.