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**Delayed feedback versus seasonal forcing: resonance phenomena in an El Niño Southern Oscillation model.**
*(English)*
Zbl 1320.37038

Summary: Climate models can take many different forms, from very detailed highly computational models with hundreds of thousands of variables, to more phenomenological models of only a few variables that are designed to investigate fundamental relationships in the climate system. Important ingredients in these models are the periodic forcing by the seasons, as well as global transport phenomena of quantities such as air or ocean temperature and salinity. We consider a phenomenological model for the El Niño Southern Oscillation system, where the delayed effects of oceanic waves are incorporated explicitly into the model. This gives a description by a delay differential equation, which models underlying fundamental processes of the interaction between internal delay-induced oscillations and the external forcing. The combination of delay and forcing in differential equations has also found application in other fields, such as ecology and gene networks. Specifically, we present exemplary stable solutions of the model and illustrate bistability in the form of one-parameter bifurcation diagrams for the seasonal forcing strength parameter. So-called maximum maps are calculated to show regions of bistability in a two-parameter plane for the seasonal forcing strength and oceanic wave delay time. To explain the observed solutions and their multistabilities, we conduct a bifurcation analysis of the model by means of dedicated continuation software. Knowing for which parameter values certain bifurcations take place allows us to explain and expand on some features of the model found in previous publications concerning the existence of unstable solutions, multistability, and chaos. We uncover surprisingly complicated behavior involving the interplay between seasonal forcing and delay-induced dynamics. Resonance tongues are found to be a prominent feature in the bifurcation diagrams and they are responsible for a high degree of multistability in the model. We find bistability within certain resonance tongues as a result of a symmetry property of the governing delay differential equation. We investigate the coexistence of stable tori, how they relate to each other, and bifurcate, which involves bifurcations of invariant tori.

### MSC:

37N10 | Dynamical systems in fluid mechanics, oceanography and meteorology |

34K18 | Bifurcation theory of functional-differential equations |

37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |

37M20 | Computational methods for bifurcation problems in dynamical systems |

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\textit{A. Keane} et al., SIAM J. Appl. Dyn. Syst. 14, No. 3, 1229--1257 (2015; Zbl 1320.37038)

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