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.

Hardware security for an Internet of Things (IoT) or cyber physical system drives the need for ubiquitous cryptography to different sensing infrastructures in these fields. In particular, generating strong cryptographic keys on such resourceconstrained device depends on a lightweight and cryptographically secure random number generator. In this research work, we have introduced a new hardware chaos-based pseudorandom number generator, which is mainly based on the deletion of an Hamilton cycle within the N-cube (or on the vectorial negation), plus one single permutation. We have rigorously proven the chaotic behavior and cryptographically secure property of the whole proposal: the mid-term effects of a slight modification of the seed (proven to be sensitive to the initial conditions) or of the inputted generator cannot be predicted. The proposal has been fully deployed on a FPGA and 65nm ASIC, it runs completely in parallel while consuming as low resources as possible, and achieving: (a) 11.5 Gbps for FPGA and 9.4 Gbps for ASIC random bit throughput, (b) 3.3µW (LF) to 7.8mW (UHF) total power consumption with 5% leakage power, measured at 1.32V , and (c) able to successfully pass the statistical tests of NIST and TestU01 (BigCrush).