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

TL;DR

This paper provides a theoretical analysis of stability pockets in a periodically forced oscillator model for seasonality, complementing previous numerical studies by constructing the Poincaré map and explaining the formation of Arnol'd tongues.

Contribution

It introduces a theoretical framework for understanding stability pockets in seasonality models by analyzing the Poincaré map and the structure of Arnol'd tongues with folds.

Findings

Stability pockets correspond to Arnol'd tongues with folds in the parameter space.

Theoretical construction of the Poincaré map explains the formation of stability pockets.

Historical context of stability pockets dating back to 1928 is discussed.

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.