Force an oscillator at a frequency near its own, and it locks. The oscillator abandons its natural rhythm and matches the drive exactly — 1:1 resonance. This is the only resonance available to the standard Kuramoto model, the canonical model of coupled oscillators. No matter how you vary the forcing amplitude or detuning, the system either locks 1:1 or doesn't lock at all. The Arnold tongue — the wedge-shaped region in parameter space where locking occurs — is a single tongue. One resonance, one tongue, one possible relationship between drive and response.
Costa and de Aguiar (arXiv:2603.04207, March 2026) show that generalizing the coupling from a scalar to a matrix opens the entire hierarchy. In the matrix-coupled Kuramoto model, oscillators are unit vectors, and the coupling between them is a constant matrix rather than a single number. When this system is periodically forced, multiple Arnold tongues appear — resonances at different rational frequency ratios, nested inside each other, forming the characteristic devil's staircase of mode-locking.
The mechanism is structural. In the scalar-coupled model, the coupling is too simple to sustain subharmonic resonance. The order parameter responds to the drive at a single frequency — the drive frequency — and the only stable phase relationship is exact matching. In the matrix-coupled model, the coupling mixes components. The order parameter has internal structure — multiple degrees of freedom that can respond at different harmonics. The matrix coupling creates routes for energy to flow between modes, enabling the system to lock at frequency ratios that the scalar model cannot access.
The result is that the complexity of the resonance landscape is set by the complexity of the coupling, not by the complexity of the oscillators. The individual oscillators are unchanged — they are still simple phase variables with a natural frequency. What changed is how they talk to each other. The richer coupling produces richer locking, not because the components became more capable, but because the connection between them became more structured.
Costa and de Aguiar, "Arnold tongues in the forced Kuramoto model with matrix coupling," arXiv:2603.04207 (March 2026).