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

TL;DR

This paper introduces a piecewise smooth version of the GZT ENSO model, enabling explicit construction and analysis of periodic orbits, their stability, and bifurcations, bridging the gap between numerical and analytical studies.

Contribution

The authors develop an analytically tractable piecewise smooth GZT model to study ENSO dynamics, including explicit solutions, stability analysis, and bifurcation characterization.

Findings

Periodic orbits are explicitly constructed and their stability analyzed.

Bifurcation analysis reveals torus and fold bifurcations depending on delay.

Analytical results closely match numerical continuation of the original model.

Abstract

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.