Arnold tongues and stochastic masking in an autonomously embedded ENSO recharge oscillator

The El Niño-Southern Oscillation (ENSO) displays a characteristic combination of seasonal phase locking, interannual variability and irregular event-to-event changes in amplitude and return time. We study these features in a seasonally forced nonlinear recharge oscillator from the viewpoint of bifurcation theory of periodic orbits. The explicitly time-periodic growth-rate modulation is transformed into an autonomous system by embedding the annual forcing into a supercritical Hopf oscillator. This formulation makes the model directly accessible to numerical continuation and Floquet analysis. We show that the annual cycle creates a resonance skeleton organized by Neimark-Sacker bifurcations, resonance points that are branch points of limit cycles and limit-point-of-cycle boundaries of Arnold tongues. The linear recharge-discharge frequency controls the location of the 1:n resonance roots, while the forcing amplitude determines the width and robustness of the corresponding tongues. Locked solutions appear as closed periodic orbits on the torus, whereas parameter values outside the Arnold tongues produce quasiperiodic motion. Slow drift of the recharge-discharge frequency causes transient capture into and escape from distinct 1:n regimes, and state-dependent stochastic forcing masks the deterministic periodic backbone by perturbing trajectories away from locked cycles. The resulting framework interprets irregular ENSO-like variability not as the absence of structure, but as a noise-perturbed system moving through a structured resonance landscape. A comparison with the Relative Oceanic Niño Index is used as an illustrative observational signature of transient phase locking rather than as a statistical attribution claim.