Delayed Feedback Versus Seasonal Forcing: Resonance Phenomena in an El Nin͂o Southern Oscillation Model

Abstract

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.

Border-collision bifurcations in a driven time-delay system

“…Additionally, the smooth system contains period doubling bifurcations, which we do not observe in our system, likely due to the loss of nonlinearity in the limit of κ → ∞. However, the analysis presented here does provide new insights into dynamics previously observed numerically 16,17 .…”

“…We label solutions by the characteristic ratio P :R, where R is the number of times the trajectory crosses from x < 0 to x > 0 in one cycle. Note that the P : R notation differs from the p : q notation used in some previous literature 17,18 , where p q is the rotation number. We find that P : R is a useful measure of a solution, for reasons that will be made clear later.…”

“…The combination of external forcing and time delay has been studied, for example, in the context of the Duffing oscillator 14 , the van der Pol oscillator 15 and more recently in the context of climate systems 16,17 . However, a general understanding of this class of dynamical systems is not yet available.…”

Enhancing noise-induced switching times in systems with distributed delays

“…In this section we perform numerical simulations of the system (1) with uniform and two-peak distribution kernels, given by (19) in order to compare theoretical predictions for switching times. Following the same methodology as in [41], we calculate the mean switching times T st as the noise η(t) takes the system out of the basin of attraction of the steady state x A to x < x S .…”

“…In addition to intrinsic time delays, majority of real systems are also affected by both external and internal random perturbations, which necessitate the inclusion of noise into the dynamical models, and hence, stochastic differential delay equations (DDEs) are used to analyze a wide range of systems. For instance, gene regulatory networks are often modeled as stochastic birth-death processes with time delays [12], and anomalies during El Niño have been analyzed using delayed-oscillator models, excited by external weather noise [17][18][19]. Recently, time-delayed feedback control has been applied to noise-induced chimera states in FitzHugh-Nagumo networks with non-local coupling, and it has been shown that for some values of time delays, it can lead to the so-called period-two coherence resonance chimera [20].…”