A beginnner's guide to limit cycles, their uses and abuses

Abstract

Mathematical models of oscillators fall into two major categories, simple (one‐dimensional) and non‐simple (two‐or‐more dimensional). The type of model used to describe a rhythmic system will influence experimental design and interpretation, and very different experimental predictions are made by simple and non‐simple models. The basic properties of non‐simple oscillators that are not shared with simple oscillators are Type 0 resetting, phase singularities, and amplitude changes. Two examples of non‐simple oscillators are adjustable‐amplitude oscillators (such as the frictionless pendulum) and attracting limit cycles. Populations of oscillators may exhibit a wide range of dynamic behaviour, including an adjustable amplitude or a limit cycle, depending on the nature of the individual oscillators and the coupling between them. Simple oscillator models may be appropriate to some biological systems such as developmental cycles and cell cycles, while circadian oscillators are best modelled by populations capable of both amplitude changes and stable entrainment.