Samuel Bolduc-St-Aubin

arXiv2026

Resonance structure of a periodically forced delay differential equation model for the El Niño--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.

math.DS↗

arXiv2025

Seasonal Forcing Dominated Dynamics of a piecewise smooth Ghil-Zaliapin-Thompson ENSO model

The Ghil-Zaliapin-Thompson (GZT) model, a scalar delay differential equation with periodic forcing and time-delayed feedback, captures key features of the El Nino-Southern Oscillation (ENSO) phenomenon. Numerical studies of the GZT model have revealed stable period-one orbits under strong forcing and locked, quasiperiodic, or even chaotic regimes under weaker forcing, but its analytical treatment remains challenging. To bridge this gap, we propose a piecewise smooth version of the GZT model with piecewise constant delayed feedback and continuous periodic forcing. For this piecewise smooth GZT model we explicitly construct solutions of initial value problems, and study the existence and properties of periodic orbits of period one. By studying the symmetries and possible phases of periodic solutions we are able to construct period-one solutions and the regions of parameter space in which they exist. We show that the stability of these orbits is governed by a linear mapping from which we find the Floquet multipliers for the periodic orbit and also the bifurcation curve along which these orbits lose stability. We show that for most values of the delay this occurs at a torus bifurcation, but that for small delays a fold bifurcation of period one orbits occurs. We then compare these analytical results with numerical continuation of the GZT model, showing that they align very closely.

math.DS↗