Arnold Maps with Noise: Differentiability and Non-monotonicity of the Rotation Number

Abstract

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.

Tipping points induced by parameter drift in an excitable ocean model

“…On the other hand, in the present case the intrinsic period of the ROs is determined by the system's dynamics, while the external-forcing period is being modified. But it is really the ratio of the www.nature.com/scientificreports/ external-to-internal frequency that matters, as illustrated in the highly simplified case of Arnold tongues in the standard circle map 86,87 .…”

Quadratic response of random and deterministic dynamical systems

“…An application to the smoothness of the rotation number of Arnold circle maps with additive noise is presented. In the paper [24], these findings are extended outside the diffeomorphism case and applied to an idealized model of El Niño-Southern Oscillation. General Linear response results for random systems were proved in [19] where the technical framework was adapted to stochastic differential equations and in [6], where the authors consider random compositions of expanding or non-uniformly expanding maps.…”

“…satisfies the assumptions of Section 5, for the sequence of spaces C k+1 (S 1 ) ⊂ C k (S 1 ) ⊂ C k−1 (S 1 ) ⊂ C k−2 (S 1 ); in particular linear response holds if we see the density of the stationary measure h δ ∈ C k−1 (S 1 ) and quadratic response holds if we consider h δ ∈ C k−2 (S 1 ). It is also possible to proceed as in [24] (Proposition 17) and deduce the regularity of the (almost surely constant) rotation number of this random dynamical system w.r.t the "driving frequency" a.…”

The physics of climate variability and climate change

“…In fact, the narrower a Devil's staircase step is, the less robust is it to noise perturbations, while the wider ones are the most robust. The effect of noise on the paradigmatic example of such a staircase, the standard circle map, has been examined in greater depth by Marangio et al (2019).…”