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

TL;DR

This work addresses the analytical intractability of the GZT ENSO model by introducing a piecewise-smooth delay-differential equation psGZT with a continuous seasonal forcing $h'(t)=-\mathrm{sign}(h(t-\tau))+c\cos(2\pi t)$. The authors develop explicit IVP solutions, construct symmetric 1:1-periodic orbits with closed-form profiles, and determine rigorous existence regions in parameter space using symmetry and phase analysis; stability is characterized via a linear map yielding Floquet multipliers, revealing torus and fold bifurcations as the main loss-of-stability mechanisms. They show that stable 1:1 periodic orbits prevail for large forcing $c$, while small delays lead to fold bifurcations and coexisting orbits; a torus bifurcation governs stability loss for most delays. Crucially, the stability and bifurcation structure of the psGZT model closely matches the bifurcation curves of the smooth GZT as the coupling parameter $\kappa$ increases, establishing psGZT as an accurate analytical proxy for the original model. The results provide a rigorous, tractable framework to understand ENSO-like delay dynamics and motivate extension to higher-period orbits (GZTn) and model refinements.

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.

Figures (11)

• Figure 1.1: Schematic representation of the time-delayed negative feedback loop of the DAO paradigm. A positive perturbation $h$ of the thermocline depth near the eastern boundary at time $t-\tau$ leads to a weakening of the easterly trade winds, resulting in westerly anomalies. These anomalies induce a mass adjustment in the central Pacific, where it is argued that the atmosphere-ocean coupling is the strongest. Such a process generates westward-propagating Rossby waves. Upon interacting with the western boundary, these waves become Kelvin waves, which cool the SSTs and terminate the warm phase at time $t$.
• Figure 1.1: Stability boundary B for symmetric $1{:}1$-periodic orbits with phase $\alpha\in[-\tfrac{1}{4},\tfrac{1}{4}]$, as in Figure \ref{['fig:T']}(a). The green curve $\textup{B}_{2,4}$ marks the transition from $2$-saddle to $4$-saddle orbits, and the blue curve $\textup{B}_{4,6}$ the transition from $4$-saddle to $6$-saddle orbits. The dashed black curve shows the corresponding bifurcation of period-one orbits in the smooth GZT model \ref{['eq:GZT']} for $\kappa=33$, where the orbit changes from $2$-saddle to $4$-saddle. The numbers $j$ label the unitary complex Floquet multipliers, indexed as in \ref{['eq:omegazero']} and \ref{['eq:omegaone']}.
• Figure 2.1: Solutions defined by \ref{['eq:closedformexpression']} and \ref{['eq:p(t)']} of the psGZT IVP \ref{['eq:iGZTIVP']} with $c=1$ and (a) $\tau=0.25$, (b) $\tau=0.76$, (c) $\tau=0.508$ and (d) $\tau=0.42$.
• Figure 3.1: Symmetric 1:1-periodic solutions shown as time series in (a1)-(b1) and projected onto the $(h(t),h(t-\tau))$-plane in (a2)-(b2). (a): 1:1-periodic solutions symmetrically related by Corollary \ref{['cor:sym2']}(2), with $c=2.75$, and delay $\tau=k+\theta$, $\theta=0.8$, and phase $\alpha$ in blue, and delay $\tau^*=k+\frac{3}{2}-\theta$, phase $\alpha^*=-\alpha$, in red. (b): symmetric 1:1-periodic orbit with $\tau=k+\frac{1}{4}$ and $c=1.5$, satisfying the stronger symmetry property \ref{['eq:D4']}.
• Figure 3.2: Illustration of Definition \ref{['def:cR']} and Theorem \ref{['thm:po']}. (a) The region $\mathcal{R}=\mathcal{R}_1\cup\mathcal{R}_2$ in light blue in the $(\theta, c)$-plane, with subregions: purple I and $\text{I}^*$ (smooth local extrema), II and $\text{II}^*$ (zero-line crossings of extrema), red III (non-smooth extrema at zero-line). (b,d) Periodic orbits within $\mathcal{R}$; (c,e) periodic orbits at $\mathcal{R}$'s boundary.

Theorems & Definitions (29)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof