# Arnold maps with noise: Differentiability and non-monotonicity of the rotation number

## TL;DR

This paper investigates Arnold maps with additive noise, demonstrating that the rotation number remains differentiable even in non-diffeomorphic regimes and revealing non-monotonic behavior when the nonlinearity is large.

## Contribution

It provides a rigorous analysis of the rotation number's differentiability and monotonicity properties in noisy Arnold maps across different nonlinear regimes.

## Key findings

- Rotation number is differentiable for all noise-perturbed cases where the map is mixing.
- Monotonicity of the rotation number with respect to frequency holds only when the map is a diffeomorphism.
- Non-monotonic behavior of the rotation number occurs when the nonlinearity parameter exceeds 1.

## Abstract

Arnold's standard circle maps are widely used to study the quasi-periodic route to chaos and other phenomena associated with nonlinear dynamics in the presence of two rationally unrelated periodicities. In particular, the El Nino-Southern Oscillation (ENSO) phenomenon is a crucial component of climate variability on interannual time scales and it is dominated by the seasonal cycle, on the one hand, and an intrinsic oscillatory instability with a period of a few years, on the other. The role of meteorological phenomena on much shorter time scales, such as westerly wind bursts, has also been recognized and modeled as additive noise. We consider herein Arnold maps with additive, uniformly distributed noise. When the map's nonlinear term, scaled by the parameter $\epsilon$, is sufficiently small, i.e. $\epsilon < 1$, the map is known to be a diffeomorphism and the rotation number $\rho_{\omega}$ is a differentiable function of the driving frequency $\omega$. We concentrate on the rotation number's behavior as the nonlinearity becomes large, and show rigorously that $\rho _{\omega }$ is a differentiable function of $\omega $, even for $\epsilon \geq 1$, at every point at which the noise-perturbed map is mixing. We also provide a formula for the derivative of the rotation number. The reasoning relies on linear-response theory and a computer-aided proof. In the diffeomorphism case of $\epsilon <1$, the rotation number $\rho_{\omega }$ behaves monotonically with respect to $\omega $. We show, using again a computer-aided proof, that this is not the case when $\epsilon \geq 1$ and the map is not a diffeomorphism.