  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.
