Torus folding, a phenomenon in which two smooth invariant tori approach each other and disappear, cannot be reduced to a simple saddle-node bifurcation. Near the folding point, the tori lose normal hyperbolicity and break apart, giving rise to complex dynamics organized into so-called Chenciner bubbles. This phenomenon has been observed in electrical circuits, models of cardiac arrhythmia, and particle accelerators, but the detailed bifurcation structure inside the bubble had long remained unconfirmed for delay systems. This work stands alongside studies of quantum states and noise-induced order, but it differs in that it analyzes a Chenciner bubble for the first time in an infinite-dimensional system: a delay differential equation, not a map.
The authors use the GZT model, a delay differential equation with negative feedback and periodic forcing, where the variable h describes the deviation of the thermocline depth in the eastern Pacific Ocean. For bifurcation analysis, they employ the software package DDE-Biftool, which allows continuing periodic solutions and computing their Floquet multipliers. Torus folding is studied through the 2:7 resonance tongue, within which they construct curves of saddle-node bifurcations, neutral saddles, homoclinic and heteroclinic transitions, and torus folding curves. Since some bifurcations involve tori, they are detected in two-dimensional projections related to Poincaré sections and through the computation of unstable manifolds of saddle periodic solutions.
Inside the Chenciner bubble for the 2:7 resonance, a bifurcation structure is found that agrees well with the theoretical picture proposed by Baesens and MacKay. Curves of neutral saddles, homoclinic and heteroclinic transitions, and torus folding curves are found. The difference from theory is that the torus bifurcation inside the bubble turns out to be subcritical rather than supercritical: a saddle torus is born, not a stable one. Transitions across curves of homoclinic and heteroclinic tangencies manifest as sharp changes in the traces of unstable manifolds of saddle periodic solutions. Numerical computations confirm that torus folding curves are not smooth bifurcation curves but rather envelopes of high-order resonance tongues.
This work demonstrates for the first time the complete bifurcation structure of a Chenciner bubble in a delay differential equation, extending the understanding of quasiperiodic bifurcations beyond finite-dimensional maps. Unlike neighboring studies that focus on quantum systems or noise, here a deterministic climate model with an infinite-dimensional phase space is analyzed. This confirms the universality of the Baesens-MacKay theoretical scheme and shows that subcriticality of the torus bifurcation may be typical for such systems. The results provide a basis for searching for similar structures in more complex climate models and other delay systems.
The topic could develop toward analyzing the influence of noise on the bifurcation structure of the Chenciner bubble, since real climate data are always noisy. It might also be possible to study the hierarchy of Chenciner bubbles that arise along torus folding curves and their role in multifrequency climate tipping. Investigating spatially distributed models, where averaged observables are analyzed instead of time series, is promising, as is checking whether signatures of torus breakup survive data reduction.
The results could influence the analysis of climate time series, especially in the search for precursors of abrupt transitions in systems with multiple frequencies. They are also useful for bifurcation analysis of models in ecology, neuroscience, and engineering, where delayed feedback and periodic forcing occur.
The next step could be to test for the presence of Chenciner bubbles in more realistic climate models, such as intermediate-complexity models for the thermohaline circulation. It is also important to study how spatial data averaging and the addition of noise affect the observable signatures of torus folding.
This work connects to the unresolved problem of identifying multifrequency climate tipping in real data, where so far only bifurcations of equilibria or periodic solutions have been considered. The question remains open whether torus folding can be distinguished from other types of bifurcations in noisy time series of limited length.
The 2:7 resonance tongue in the model means that exactly 2 feedback cycles occur every 7 years, and this tongue turned out to be the largest in the parameter region considered, which facilitated the numerical analysis.
The idea of abrupt climate transitions due to the interaction of multiple frequencies recalls the plot of the film 'The Day After Tomorrow', where disruption of ocean circulation leads to catastrophic changes, although in reality such transitions may be more gradual and related to quasiperiodic dynamics.