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In high dimension, stability and uniqueness of periodic orbits in nonlinear smooth systems are difficult properties to establish in general. In a previous work, we proved the existence of periodic oscillations inscribed in an invariant torus for a class of negative feedback systems in ${\mathbb{R}}^n$, where the regulation functions defining these systems are supposed to be nonlinear (and possibly nonmonotonic) in a small window and constant outside this window. Here, under some symmetry assumptions on the parameters of these models, we establish uniqueness and stability of the periodic orbit inside this invariant torus. The method used is based on the analysis of the spectrum of the monodromy matrix associated with the periodic orbit considered. Under the same assumptions, an approximation of the period of the orbit in terms of the parameters is also provided, and all results are illustrated with several examples from circadian rhythms.

A Stability Result for Periodic Solutions of Nonmonotonic Smooth Negative Feedback Systems