This paper is motivated by applications in oscillatory systems where researchers are typically interested in discovering components of those systems that display rhythmic temporal patterns. The contributions of this paper are twofold. First, a methodology is developed based on acircular signalplus error model that is defined using order restrictions. This mathematical formulation of rhythmicity is simple, easily interpretable and very flexible, with the latter property derived from the nonparametric formulation of the signal. Second, we address various commonly encountered problems in the analysis of oscillatory systems data. Specifically, we propose a methodology for (a) detecting rhythmic signals in an oscillatory system and (b) estimating the unknown sampling time that occurs when tissues are obtained from subjects whose time of death is unknown. The proposed methodology is computationally efficient, outperforms the existing methods, and is broadly applicable to address a wide range of questions related to oscillatory systems.