The devil's staircase is what mathematicians call a graph that looks like steps leading upward, but the steps are infinitely thin and packed without gaps. In the deterministic Arnold map, the simplest model of two incommensurate oscillations, the rotation number forms exactly such a staircase for weak nonlinearity: it sticks at every rational frequency value and rises only in jumps. Add a little random noise to each step, and the steps begin to smooth out, turning into a gentle curve.
But what happens when the nonlinearity becomes strong? Until now, that remained an open question. In the new work, the authors consider a stochastic Arnold map where, at each step, an independent uniformly distributed noise is added to the deterministic shift. They prove that even for nonlinearity ε ≥ 1, when the map ceases to be a diffeomorphism, the rotation number remains a differentiable function of the frequency parameter τ at all points where the system is mixing. In other words, noise continues to polish the staircase, but now it can bend.
However, smoothing does not mean preserving all previous properties. For ε = 1.4 and a small noise amplitude ξ = 0.01, the authors found that the rotation number ceases to be monotonic: as τ increases from 0.707 to 0.716, it first decreases and then increases. The staircase, which previously only went up, now takes a step down. For sufficiently large noise, for example ξ = 0.05, this nonmonotonicity disappears, and the curve becomes smooth and unidirectional again.
This result connects linear response theory with climate models. In the El Niño phenomenon, the seasonal cycle and internal ocean instability interact, and short-period disturbances such as westerly wind bursts are modeled as additive noise. If the rotation number can depend nonmonotonically on frequency, then frequency locking can occur for disconnected parameter regions, complicating the prediction of climate oscillations.
This work is the first to rigorously establish differentiability of the rotation number for strong nonlinearity in the presence of noise. Previously, this was known only for diffeomorphisms, that is, when ε < 1. The discovered nonmonotonicity points to new phenomena in random dynamical systems and raises the question of how noise simultaneously restores smoothness and generates unexpected bends in the staircase. The methods of computer-assisted estimates can be extended to other families of maps and to more realistic climate models, where noise is already used to parameterize subgrid processes.
The name "devil's staircase" for the graph of the rotation number reflects its step-like structure: at every rational frequency value the rotation number is constant, and between steps there are jumps. Adding noise "smooths" this staircase, turning it into a smooth curve, but under strong nonlinearity the staircase can bend downward, breaking monotonicity.