El Niño–Southern Oscillation (ENSO) is the largest source of climate variability on the planet, affecting weather from Australia to the Andes. The classic Suarez–Schopf model, proposed in 1988, describes the interaction of Rossby and Kelvin waves with a single scalar delay equation. For a long time it was thought that oscillations arise only under linear instability of equilibria, but the authors show that complex regimes hide in the stability zone. This work stands alongside studies of blazars, gravitational waves, and quantum clocks, but stands out for its focus on climate dynamics and hidden attractors.
The authors use an analytical–numerical approach. Using inertial manifold theory developed in their previous works, they prove the existence of a two-dimensional manifold that attracts all trajectories. This allows applying two-dimensional Poincaré–Bendixson arguments. For numerical integration, they use the JiTCDDE package in Python with a step of 1e-3 and accuracy 1e-5. Hidden orbits are located by linear feedback with parameter continuation, and chaos under periodic forcing is studied via Lyapunov exponents.
In the region of linear stability, two subregions are found: self-excited and hidden periodic orbits. For α=0.75 and τ=1.65, the self-excited orbit has a period of about 12.3 (8.17 years), while for τ=1.58 the hidden orbit has a period of 15.3 (10.6 years). Between them lies a homoclinic figure-eight that bifurcates into one or two unstable orbits. Under small periodic forcing (amplitude A=0.073), chaos is observed with a positive Lyapunov exponent λ1≈0.079. Hidden and self-excited asynchronous regimes are also found in a ring of two coupled transmission lines.
The work shows that El Niño irregularity can arise not only from external noise or chaos, but also from multistability within the deterministic model itself. This changes the view on forecasting: even without random forcing, the system can jump between regimes. Unlike neighboring works, here the existence of an inertial manifold and hidden attractors is rigorously established for the first time for a climate model with delay, opening the way to analytical description.
The topic could develop toward proving the conjecture about the absence of oscillations when λ1+λ2<0 and testing Eden's hypothesis on the coincidence of local and global Lyapunov dimensions. The methods may extend to more complex ENSO models with multiple delays and spatial structure.
The results will affect climate modeling, the theory of delay differential equations, and the analysis of nonlinear networks.
To construct more precise estimates of the dissipative region and prove the existence of homoclinic curves using normal form theory.
The work links the unsolved problem of ENSO irregularity to fundamental questions in dynamical systems theory: the existence of hidden attractors, bifurcations of homoclinic orbits, and the transition to chaos in delay systems.
The period of hidden oscillations in the model can vary from 3.85 to 5.37 years depending on parameters, which covers the observed range of real El Niño events.