The Arnold map serves as the simplest model of the interaction between two incommensurate oscillations, such as the seasonal cycle and internal instability in the climate system. In the deterministic case, for weak nonlinearity, the rotation number forms a "devil's staircase": it is constant on frequency-locking intervals and increases monotonically. However, describing El Niño's irregularity requires strong nonlinearity and the inclusion of short-period perturbations, such as westerly wind bursts, which are modeled as additive noise. This work sits alongside studies of turbulence and quantum chaos, where noise is also considered a regularizing factor, but here the case of strong nonlinearity in the Arnold map is rigorously investigated for the first time.
The authors consider a stochastic Arnold map on the circle, where at each step an independent uniformly distributed noise is added to the deterministic shift. To prove differentiability of the rotation number, they use linear response theory: they show that the stationary measure depends smoothly on the frequency parameter if the transfer operator is mixing. Mixing is verified via computer-assisted estimates that approximate the transfer operator by a finite-dimensional operator and provide rigorous bounds on the rate of convergence to equilibrium. Non-monotonicity of the rotation number is established by certified computation of the stationary measure with a small error in the L1 norm for specific parameter values.
It is proven that for nonlinearity ε ≥ 1, the rotation number ρτ is a differentiable function of the parameter τ at every point where the stochastic system is mixing. An explicit formula for the derivative is obtained via the resolvent of the transfer operator and the derivative of the stationary measure. Computer-assisted estimates confirm mixing for ε = 1.4, ξ = 0.1, and τ ∈ [0.75, 0.8], which yields differentiability on this interval. For ε = 1.4 and ξ = 0.01, the rotation number is shown to be non-monotonic: as τ increases from 0.707 to 0.716, it first decreases and then increases. Numerical experiments indicate that non-monotonicity disappears for sufficiently large noise (e.g., ξ = 0.05).
This work differs from neighboring studies in that it is the first to rigorously establish differentiability of the rotation number for strong nonlinearity in the presence of noise, whereas previously this was known only for diffeomorphisms (ε < 1). Furthermore, the discovered non-monotonicity of the rotation number for ε ≥ 1 contrasts with the monotonic behavior in the weakly nonlinear case and points to the possibility of frequency locking for disconnected parameter regions. This expands the understanding of noise as a regularizer in random dynamical systems and connects linear response theory with climate models.
The methods of computer-assisted estimates can be extended to other families of maps with noise, including systems with Gaussian noise or more complex nonlinearities. Application to intermediate-complexity climate models, where additive noise is already used to parameterize subgrid processes, is promising. It is also interesting to investigate how non-monotonicity of the rotation number affects the predictability of El Niño in models with a hierarchy of complexity.
The results may influence the modeling of El Niño and other climate oscillations, as well as the analysis of synchronization in biological systems such as cardiac arrhythmias and neural rhythms.
The next step could be extending the differentiability proof to the case of variable noise amplitude and to maps with more general nonlinearities. It is also important to study under what conditions non-monotonicity of the rotation number arises in realistic ENSO models.
In dynamical systems theory, the question of smoothness of statistical properties for systems with strong nonlinearity without noise remains open. This work shows that noise can restore smoothness, but at the same time it generates new phenomena, such as non-monotonicity of the rotation number, which is related to the problem of describing the transition to chaos in random systems.
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.