The El Niño–Southern Oscillation (ENSO) is the most important climate phenomenon affecting weather worldwide. Its phases, El Niño and La Niña, are characterized by sea surface temperature anomalies in the equatorial Pacific and have profound impacts on global climate. Observations show that these events have a pronounced seasonal timing: peaks typically occur in December. Early explanations, such as the Bjerknes positive feedback, described the mechanism amplifying anomalies but could not explain their reversal. Conceptual models, including the delayed action oscillator (DAO), were proposed to formalize the interplay of positive and negative feedbacks. This work focuses on the DAO model proposed by Suárez and Schopf, which has recently gained physical justification through reduction of spatially distributed models. Unlike more complex models, it is transparent enough for detailed mathematical analysis, allowing the study of resonance phenomena under periodic forcing.
The authors use a combination of analytical and numerical methods. For the autonomous Suárez–Schopf model (without external forcing), they perform a linear stability analysis of equilibria, obtaining analytical expressions for pitchfork and Hopf bifurcations. To find bifurcations not captured by linear analysis (e.g., saddle-node bifurcations of periodic orbits and homoclinic bifurcations), they use the continuation software DDE-BifTool. For the periodically forced model, the authors develop a method to compute the rotation number on attracting invariant tori. This method generalizes an approach previously proposed for ordinary differential equations to the infinite-dimensional phase space of delay systems. The key idea is to iterate the stroboscopic map until the trajectory returns to a small neighborhood of the initial point, after which the rotation number is computed as the average angular displacement. To overcome projection issues in infinite-dimensional space, two different projections onto a plane are used, and the one that yields a continuous dependence of the rotation number on parameters is selected.
For the autonomous Suárez–Schopf model, a complete bifurcation diagram is constructed in the parameter plane (delay τ, feedback strength α). It is shown that for α > 1, the only attractor is a symmetric limit cycle arising from a supercritical Hopf bifurcation. For α < 1, a more complex picture emerges: after a pitchfork bifurcation, two symmetric equilibria appear, which then lose stability via a subcritical Hopf bifurcation; meanwhile, between regions with different attractors, a saddle-node bifurcation of periodic orbits and a homoclinic gluing bifurcation occur. For the periodically forced model (with frequency ω = 2π), regions with an attracting invariant torus are found in the (τ, c) plane (c is the forcing amplitude). Resonance tongues (phase synchronization regions) turn out to be very narrow and do not overlap, indicating weak synchronization. The structure of resonance tongues is organized by critical points of the rotation number graph both on torus bifurcation curves and inside the torus existence region. At large delays, the resonance structure repeats with period 1 in τ, described by the rational map Φ(p:q) = p:(q+2p). For α = 0.75, bistability between period-1 periodic orbits and invariant tori is found, involving nonclassical 'fold of tori' and 'gluing of tori' bifurcations.
This work significantly advances the understanding of periodically forced delay systems, which are widely used in climate modeling. The developed method for computing rotation numbers in delay systems opens new avenues for studying resonance phenomena in such systems. The discovered repetition mechanism of the resonance structure at large delays, described by a transformation based on a matrix from SL(2,Z), is novel and may be universal across a wide class of models. From a climatological perspective, the results show that within the Suárez–Schopf model, seasonal forcing leads only to weak phase synchronization, consistent with the observed irregularity of ENSO. However, the lack of overlap of resonance tongues means the model does not reproduce the chaotic dynamics seen in real data, indicating the need for further model complexity.
The topic can develop in several directions. First, it is of interest to investigate the dependence of the resonance structure on the forcing frequency ω, since changes in ω may lead to overlapping resonance tongues and chaotic dynamics. Second, one could consider breaking the Z2 symmetry of the model to account for the asymmetry between El Niño and La Niña. Third, including stochastic forcing could enhance irregularity and bring the model closer to real data. Finally, studying other conceptual ENSO models, such as the Gil–Zaliapin–Thompson model, using the developed methods may shed light on general patterns of resonance phenomena in climate oscillators.
The results may influence the development of more realistic conceptual ENSO models used for forecasting climate anomalies. The rotation number method is applicable to a wide range of periodically forced problems in physics, biology, and engineering.
The immediate next step is a systematic study of the influence of forcing frequency ω on the resonance structure and the onset of chaos. It is also important to compare with other ENSO models and observational data to assess model adequacy.
This work directly addresses the fundamental problem of understanding the mechanisms of ENSO irregularity. Existing conceptual models, including the one considered, do not reproduce the observed level of chaos, indicating the need to incorporate additional factors such as phase asymmetry, stochastic forcing, or more complex nonlinearities. Studying resonance phenomena in such models helps identify conditions under which a transition to chaos is possible, thereby bringing us closer to explaining the real variability of the climate system.
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What the work does
This work examines the resonant structure of the Suarez–Schopf equation with delay under periodic forcing. The authors constructed a bifurcation diagram for the autonomous model, developed a method for computing the rotation number in delay systems, and found that resonance tongues repeat for large delays.
Where its strength is
The main strength is the development and application of a method to compute the rotation number on invariant tori for infinite-dimensional delay systems, which allows a detailed study of resonant phenomena.
Where it could go next
Worth trying
Why this may matter
Understanding resonant phenomena in climate models could improve predictions of El Niño phases and their seasonal timing, which is important for assessing global climate risks.
Nearby works in our archive
The archive is sparse around this work: the closest review is only 0.49, against a usual 0.64. It looks like few have drilled nearby.
The rotation number on an invariant torus can be rational (then the dynamics are periodic) or irrational (then the trajectory densely fills the torus). In this work, computing the rotation number required about 768,000 calculations, taking roughly 8 hours on an Apple M1 computer.