The GZT model describes oscillations in the depth of the thermocline (the layer of sharp temperature change) in the eastern tropical Pacific. It embodies the idea of a delayed negative oscillator: Kelvin and Rossby waves propagating across the ocean create feedback with a delay. Numerical studies have revealed rich dynamics, including periodic, quasiperiodic, and chaotic regimes, but rigorous analysis remained a challenge. This work stands alongside studies of other conceptual climate models with delay, such as the Suarez–Schopf model and the Timmermann–Stone–Kane–Jarosch model, as well as recent works on piecewise-smooth systems with delay.
The authors considered the limiting case of the GZT model for large coupling parameter, where the hyperbolic tangent is replaced by a sign function. The resulting delay equation has piecewise-constant feedback and smooth periodic forcing. For such a system, an explicit solution to the Cauchy problem was constructed using the method of steps: on each interval of length equal to the delay, the equation reduces to an ordinary differential equation. Then symmetries of periodic solutions were studied, and orbits of period one with one zero crossing per period were constructed. Stability of orbits was determined via the evolution of zeros of a perturbed solution, yielding a characteristic polynomial and Floquet multipliers.
Explicit expressions for period-one periodic orbits were derived as functions of model parameters. Regions of existence of such orbits in the (delay, forcing amplitude) plane were determined. It was shown that for large forcing amplitudes the orbits are stable, and loss of stability occurs either through a torus bifurcation or a fold. Closed-form formulas for bifurcation curves and rotation numbers were obtained. Comparison with numerical continuation of the original GZT model showed excellent agreement: the bifurcation curves and rotation numbers nearly coincide, especially for large values of the coupling parameter and delay. Unlike a previously studied model with piecewise-constant forcing, the new model reproduces a smooth dependence of stability on parameters.
This work demonstrates that the piecewise-smooth approximation with smooth seasonal forcing is a much more accurate analytical proxy for the GZT model than the model with piecewise-constant forcing. It allows a rigorous explanation of bifurcation mechanisms in the original model that were previously observed only numerically. In contrast to neighboring works that used either fully numerical methods or overly crude simplifications, this study strikes a balance between analytical tractability and fidelity to the dynamics. The results provide a new tool for understanding delayed oscillators in climate models.
In the future, the analysis could be extended to orbits of higher periods and the mechanisms of frequency synchronization could be studied. It would also be interesting to add asymmetry in the coupling function, positive feedback, and state-dependent delay, which would bring the model closer to real ENSO data. Such extensions, as studies of the smooth GZT model show, lead to more complex and chaotic dynamics.
The results may be useful in climatology for interpreting ENSO regimes and their bifurcations under parameter changes. They also contribute to the general theory of delay differential equations and piecewise-smooth dynamical systems.
The next step will be to study orbits with more zero crossings and resonance tongues. A comparison with other conceptual ENSO models and analysis of the influence of additional physical factors are also planned.
This work is connected to the unsolved problem of explaining the complex dynamics of ENSO, including irregularity and amplitude variability. It clarifies how delayed feedback and seasonal forcing interact to produce bifurcations, which may help in understanding observed interannual and interdecadal climate variability.
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What the work does
The authors constructed a piecewise-smooth version of the Gil–Zaliapin–Thompson model, replacing the hyperbolic tangent with a sign function while keeping the smooth seasonal forcing. For this system, explicit solutions were obtained using the method of steps, periodic orbits of period one were found, and their stability was analytically investigated through the evolution of zeros. Comparison with numerical continuation of the original model showed excellent agreement, especially for large parameter values.
Where its strength is
This work provides a rigorous analytical description of the stability bifurcations of periodic orbits in the GZT model, which was previously only accessible numerically. Explicit formulas for the bifurcation curves and the rotation number allow one to accurately reproduce the dynamics of the original model over a wide parameter range.
Where it could go next
Worth trying
Why this may matter
The analytical tractability of the piecewise-smooth model opens the way to a rigorous study of the mechanisms governing the complex dynamics of ENSO, which may improve understanding and possibly prediction of this climate phenomenon.
Nearby works in our archive
The archive is sparse around this work: the closest review is only 0.51, against a usual 0.64. It looks like few have drilled nearby.
The GZT model was proposed in 2008 and named after three scientists: Michael Ghil, Ilya Zaliapin, and Michael Thompson. It has become one of the most studied conceptual ENSO models due to its simplicity and ability to reproduce key features of the phenomenon.