Chenciner bubbles and torus break-up in a periodically forced delay differential equation

TL;DR

This paper investigates the complex bifurcation structures, specifically Chenciner bubbles, in a delay differential equation model of climate dynamics, revealing how tori break up and influence system tipping points.

Contribution

First analysis of Chenciner bubbles in a delay differential equation model, linking bifurcation theory with climate system dynamics and tori breakdown.

Findings

Identification of a Chenciner bubble in a delay differential equation model.

Bifurcation analysis showing agreement with theoretical predictions.

Insights into the role of folding tori in climate tipping points.

Abstract

We study a generic model for the interaction of negative delayed feedback and periodic forcing that was first introduced by Ghil et al. in the context of the El Ni\~no Southern Oscillation (ENSO) climate system. This model takes the form of a delay differential equation and has been shown in previous work to be capable of producing complicated dynamics, which is organised by resonances between the external forcing and dynamics induced by feedback. For certain parameter values, we observe in simulations the sudden disappearance of (two-frequency dynamics on) tori. This can be explained by the folding of invariant tori and their associated resonance tongues. It is known that two smooth tori cannot simply meet and merge; they must actually break up in complicated bifurcation scenarios that are organised within so-called resonance bubbles first studied by Chenciner. We identify and analyse such a Chenciner bubble in order to understand the dynamics at folds of tori. We conduct a bifurcation analysis of the Chenciner bubble by means of continuation software and dedicated simulations, whereby some bifurcations involve tori and are detected in appropriate two-dimensional projections associated with Poincar\'e sections. We find close agreement between the observed bifurcation structure in the Chenciner bubble and a previously suggested theoretical picture. As far as we are aware, this is the first time the bifurcation structure associated with a Chenciner bubble has been analysed in a delay differential equation and, in fact, for a flow rather than an explicit map. Following our analysis, we briefly discuss the possible role of folding tori and Chenciner bubbles in the context of tipping.