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

A periodically forced oscillator in a model for seasonality shows chains of stability pockets in the parameter plane. The frequency of the oscillator and the length of the photoperiod in the Zeitgeber are the two parameters. The present study is intended as a theoretical complement to the numerical study of Schmal et al. (2015) of stability pockets (or Arnol'd onions in their terminology). We construct the Poincaré map of the forced oscillator and show that the Arnol'd tongues are taken into chains of stability pockets by a map with a number of folds. This number is related to the rational point (pq,0) on the frequency axis from which a chain of p pockets emanates. Stability pockets are already observed in an article by van der Pol and Strutt in 1928, see van der Pol and Strutt (1928) and later explained by Broer and Levi in 1995, see Broer and Levi (1995).