Stability pockets of a periodically forced oscillator in a model for seasonality

Abstract

A periodically forced oscillator in a model for seasonality shows stability pockets and chains thereof in the parameter plane. The frequency of the oscillator and the season indicated by a value between zero and one are the two parameters. The present study is intended as a theoretical complement to the numerical study of Schmal et al. in 2015 of stability pockets or Arnol'd onions in their terminology. We construct the Poincar\'e map of the forced oscillator and show that the Arnol'd tongues are taken into stability pockets by a map with a number of folds. Stability pockets are already observed in an article by van der Pol \& Strutt in 1928 and later explained by Broer \& Levi in 1995.