We show a linear response statement for fixed points of a family of Markov operators, which are perturbations of mixing and regularizing operators. We apply the statement to random dynamical systems on the interval given by a deterministic map T with additive noise (distributed according to a bounded variation kernel). We prove linear response for these systems, also providing explicit formulas both for deterministic perturbations of the map T and for changes in the noise kernel. The response holds with mild assumptions on the system, allowing the map T to have critical points, contracting and expanding regions. We apply our theory to topological mixing maps with additive noise, to a model of the Belozuv–Zhabotinsky chemical reaction and to random rotations. In the final part of the paper we discuss the linear request problem for these kind of systems, determining which perturbations of T produce a prescribed response.

How can one change a system, in order to change its statistical properties in a prescribed way? In this note we consider a control problem related to the theory of linear response. Given an expanding map of the unit circle with an associated invariant density, we can consider the inverse problem of finding which first order changes in the transformation can achieve a given first order perturbation in the density. We show the general mathematical structure of the problem, the existence of many solutions in the case of expanding maps of the circle and the existence of optimal ones. We investigate in depth the example of the doubling map, where we give a complete solution of the problem.

We consider the linear and quadratic higher-order terms associated with 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 changes in the system itself, expressing how the statistical properties of the system vary 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.

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 resource-constrained 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) 3.3μ W (LF) to 7.8 mW (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).

We present a general setting in which the formula describing the linear response of the physical measure of a perturbed system can be obtained. In this general setting we obtain an algorithm to rigorously compute the linear response. We apply our results to expanding circle maps. In particular, we present examples where we compute, up to a pre-specified error in the L ∞ -norm, the response of expanding circle maps under stochastic and deterministic perturbations. Moreover, we present an example where we compute, up to a pre-specified error in the L ¹ -norm, the response of the intermittent family at the boundary; i.e. when the unperturbed system is the doubling map.