Resonance structure of a periodically forced delay differential equation model for the El Ni\~no--Southern Oscillation

We study resonance phenomena in the periodically forced Suarez--Schopf delay differential equation, which is a conceptual climate model for the El Niño--Southern Oscillation (ENSO).

The system serves as a prototypical forced delayed-action oscillator whose self-sustained oscillations, when subjected to periodic forcing, give rise to attracting invariant tori.

We provide a comprehensive bifurcation analysis of both the unforced and the forced model; for the latter, we propose a method to compute the rotation number of normally hyperbolic attracting invariant tori.

With it we show that resonance tongues in parameter space are organized by critical points of the graph of the rotation number, both along torus bifurcation curves and within the region of invariant tori.

We also show that the resonance structure repeats for large delays, which constitutes a reappearance mechanism not previously reported in the literature.

Furthermore, depending on the feedback strength, we find bistability between period-one orbits and invariant tori.

This regime involves non-classical bifurcation sequences, including `saddle-node' and `gluing' bifurcations of tori.