Coexistence of attractors in a coupled nonlinear delayed system modelling El Niño Southern Oscillations

Abstract: We study the dynamics of the sea surface temperature (SST) anomaly using a model of the temporal patterns of two sub-regions, mimicking behaviour similar to El Niño Southern Oscillations (ENSO). Specifically, we present the existence, stability, and basins of attraction of the solutions arising in the model system in the space of these parameters: self delay, delay and inter-region coupling strengths. The emergence or suppression of oscillations in our models is a dynamical feature of utmost relevance, as it signals the presence or absence of ENSO-like oscillations. In contrast to the well-known low order model of ENSO, where the influence of the neighbouring regions on the region of interest is modelled as external noise, we consider neighbouring regions as a coupled deterministic dynamical systems. Different parameters yield a rich variety of dynamical patterns in our model, ranging from steady states and homogeneous oscillations to irregular oscillations and coexistence of oscillatory attractors, without explicit inclusion of noise. Interestingly, if we take the self-delay coupling strengths of the two sub-regions to be such that the temperature of one region goes to a fixed point regime when uncoupled, while the other system is in the oscillatory regime, then on coupling both systems show oscillations. We explicitly obtain the basins of attraction for the different steady states and oscillatory states in the model. Our results might be helpful for forecasting of El Niño (or La Niña) progress, as it indicates the combination of initial SST anomalies in the sub-regions that can result in a El Niño/La Niña episodes. In particular, the result suggests using an interval as a criterion to estimate the El-Niño or La-Niña progress instead of the currently used the single value criterion.

• The paper demonstrates that a coupled nonlinear delayed oscillator model can produce multiple coexisting attractors—steady states and oscillatory regimes—relevant to ENSO phenomena.
• It utilizes symmetric delay differential equations to map bifurcation patterns, revealing how self-delay and inter-region coupling modulate SST anomaly oscillations.
• The results indicate that deterministic inter-region interactions can both induce and suppress ENSO oscillations, offering insights for probabilistic forecasting of oceanic regimes.

Coexistence of Attractors in a Coupled Nonlinear Delayed System Modeling ENSO

Introduction

This work presents an analysis of the dynamical regimes in a coupled nonlinear delayed system formulated to model El Niño–Southern Oscillation (ENSO) phenomena. The system extends classical delayed oscillator frameworks by considering two deterministically coupled sub-regions, each modeled by a nonlinear first-order delay differential equation (DDE) for sea surface temperature (SST) anomaly. The approach departs from previous models that treated neighboring region influences as stochastic inputs, instead encoding mutual determinism via inter-region coupling. The study systematically characterizes the emergence, stability, coexistence, and basins of attraction for distinct oscillatory and steady-state solutions as functions of self-delay, delay time, and inter-region coupling strengths.

Model Formulation

The dimensionless model comprises two coupled sub-units:

$$\frac{dT_{1}}{dt} = T_{1} - T_{1}^{3} - \alpha_{1} T_{1}(t - \delta_{1}) + \gamma T_{2}$$

$$\frac{dT_{2}}{dt} = T_{2} - T_{2}^{3} - \alpha_{2} T_{2}(t - \delta_{2}) + \gamma T_{1}$$

where $T_i$ are the scaled SST anomalies, $\alpha_i$ and $\delta_i$ are strengths and timescales of self-delay feedback in each sub-region, and $\gamma$ denotes symmetric linear coupling between sub-regions. This minimal deterministic model encapsulates essential feedbacks (positive, nonlinear damping, and delayed negative) implicated in ENSO dynamics while allowing for spatial heterogeneity and interaction.

Dynamical Regimes and Bifurcations

Identical Sub-regions

For parameters with $\alpha_1 = \alpha_2$, $\delta_1 = \delta_2$, the coupled DDE system exhibits distinct qualitative regimes:

• Amplitude Death (AD): Both sub-regions converge to a common steady state.
• Oscillation Death (OD): Sub-regions reach static, but distinct, fixed points.
• Homogeneous Oscillations: Synchronous oscillatory dynamics.
• Heterogeneous/Irregular Oscillations: Asynchronous or complex, possibly chaotic, oscillations.

Key bifurcation patterns are governed mainly by the delay time $\delta$ and the coupling strengths. Increased delay and self-delay coupling favor oscillatory solutions, characteristic of ENSO-like behavior, whereas strong inter-region coupling suppresses oscillations, expanding AD regions. As delay increases, oscillatory regimes emerge at lower self-delay coupling thresholds.

Non-identical Sub-regions

For $\alpha_1 \neq \alpha_2$ or $\delta_1 \neq \delta_2$, the parameter space yields richer behaviors:

• Substantial heterogeneity in self-delay yields asymmetry in amplitudes and regularity of oscillations across regions.
• Under weak coupling, dissimilar sub-regions may exhibit markedly divergent behaviors, including cases where one region displays steady-state dynamics when uncoupled but is induced to oscillate via interaction with an oscillatory region.
• Large delay disparities generate spatio-temporal complexity mimetic of observed ENSO SST anomaly records.

Multistability and Basins of Attraction

A comprehensive mapping of attractor basins reveals strong multistability. Both fixed points and oscillatory limits can coexist, with their respective basins shaped by initial conditions and parameter values. Importantly:

• Weak inter-region coupling and low delay yield multiple attractors, with intricate, often fractal-like basin boundaries.
• Increased coupling shrinks the coexistence region, favoring global convergence to AD or OD states.
• For parameter sets mapping to observed oceanic sub-region conditions, the model identifies sizable domains in phase space supporting ENSO-like oscillations, in addition to multistability between warm (El Niño), cold (La Niña), and neutral states.

These findings imply that the preconditioning of the Pacific system, as encoded by initial SST anomaly combinations in key regions, critically determines subsequent ENSO outcomes. This dynamical structure suggests that probabilistic forecasting should leverage the geometry of basin boundaries rather than rely solely on fixed thresholds for anomaly indicators.

Noise Robustness

Embedding the coupled DDE system in a stochastic framework with additive, delta-correlated Gaussian noise reveals:

• Certain attractors, particularly those associated with larger-magnitude fixed points, display strong robustness to weak and moderate noise.
• Strong noise can trigger random transitions (switching) between attractors, but rarely induces excursions to states not already observed in the deterministic system.
• Systems with weaker self-delay coupling support a greater number of robust steady states; larger noise strengths are required for state-switching in such cases.

These results highlight the likely persistence of specific SST regimes against environmental and observational fluctuations, with implications for the theoretical predictability and resilience of ENSO-like modes.

Implications for ENSO Forecasting and Theory

The analysis constrains the theoretical mechanisms responsible for the emergence and suppression of ENSO oscillations:

• Delayed negative feedback, reflecting oceanic wave transits, is indispensable for oscillatory dynamics.
• Mutual deterministic coupling alters the bifurcation landscape compared to single-site or stochastically perturbed models, both enabling and constraining multistability in the coupled Pacific system.
• The results suggest that forecasting should account for the multidimensional structure of dynamical basins. The model directly recovers the conventional $0.5^\circ$C SST anomaly threshold for ENSO identification (after appropriate scaling), but more generally indicates that intervals of initial SST anomaly values should be utilized as probabilistic criteria for regime identification.

Notably, the finding that oscillations can be induced in a 'quiescent' region via coupling to a region exhibiting oscillatory dynamics provides a mechanistic explanation for observed phase-locking and spatial propagation of ENSO events.

Future Directions

The formalism is amenable to several extensions relevant to practical and theoretical investigation:

• Incorporation of further spatial structure (more sub-regions, asymmetric coupling).
• Explicit parameterizations for seasonal forcing or slowly varying background states.
• Advanced uncertainty quantification integrating observational SST datasets for real-time regime assessments.
• Exploration of higher-dimensional attractor structures (e.g., tori or chaotic sets) under realistic parameterizations.

Such developments would refine the predictive power and realism of low-order deterministic ENSO models, bridging the gap to full GCMs while maintaining analytical and computational tractability.

Conclusion

This study provides a rigorous description of the coexistence and structure of attractors in a minimal, deterministically coupled delayed oscillator model for ENSO (1707.09853). It demonstrates that mutual interaction between sub-regions can both enhance and suppress oscillatory behavior, modulates the existence and robustness of multiple attractors, and fundamentally alters the geometry of phase space basins relevant for prediction. These results offer new theoretical insight into ENSO dynamics, indicate observable consequences in coupled SST fields, and invite the development of interval-based probabilistic forecasting criteria for ENSO phase identification.