We consider the linear and quadratic higher order terms associated to the response of the statistical properties of a dynamical system to suitable small perturbations. These terms are related to the first and second derivative of the stationary measure with respect to the change of some parameters, expressing how the statistical properties of the system varies under the perturbation. We show a general framework in which one can obtain rigorous convergence and formulas for these two terms. The framework is flexible enough to be applied both to deterministic and random systems. We give examples of such an application computing linear and quadratic response for Arnold maps with additive noise and deterministic expanding maps.

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Arnold's standard circle maps are widely used to study the quasiperiodic route to chaos and other phenomena associated with nonlinear dynamics in the presence of two rationally unrelated periodicities. In particular, the El Niño-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 , is sufficiently small, i.e. < 1, the map is known to be a diffeomorphism and the rotation number ρ ω is a differentiable function of the driving frequency ω.We concentrate on the rotation number's behavior as the nonlinearity becomes large, and show rigorously that ρ ω is a differentiable function of ω, even for ≥ 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 < 1, the rotation number ρ ω behaves monotonically with respect to ω. We show, using again a computer-aided proof, that this is not the case when ≥ 1 and the map is not a diffeomorphism. This lack of monotonicity for large nonlinearity could be of interest in some applications. For instance, when the devil's staircase ρ = ρ(ω) loses its monotonicity, frequency locking to the same periodicity could occur for non-contiguous parameter values that might even lie relatively far apart from each other.