A rigorous computational approach to linear response

Bahsoun, Wael; Galatolo, Stefano; Nisoli, Isaia; Niu, Xiaolong

doi:10.1088/1361-6544/aa9a88

Cited by 29 publications

(26 citation statements)

References 37 publications

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“…where τ x ρ := (y → ρ(y − x)) andρ(z) := ρ(−z). 5 Recall that ρ : R → R is a Schwartz function if it is a C ∞ function that satisfies, for any (n, m) ∈ N 2 ,…”

Section: 1mentioning

confidence: 99%

“…In the recent work [4] these problems are considered for general systems with additive noise and for Hilbert-Schmidt operators. Rigorous numerical approaches for the computation of the linear response are available to some extent, both for deterministic and random systems (see [5,31]). We remark that the quadratic response in principle can provide important information in these optimization problems, as can be of help in establishing convexity properties in the response of the statistical properties of a given family of systems under perturbation.…”

Section: Introductionmentioning

confidence: 99%

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Quadratic response of random and deterministic dynamical systems

Galatolo

Sedro

2020

Chaos: An Interdisciplinary Journal of Nonlinear Science

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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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“…where τ x ρ := (y → ρ(y − x)) andρ(z) := ρ(−z). 5 Recall that ρ : R → R is a Schwartz function if it is a C ∞ function that satisfies, for any (n, m) ∈ N 2 ,…”

Section: 1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quadratic response of random and deterministic dynamical systems

Galatolo

Sedro

2020

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…x) exist and jointly continuous in 3 A common way to have a parameter dependent random system is also when the system consists of a fixed probability space (Ω, P) and a parametrized family of maps Tω,ε, ω ∈ Ω, ε ∈ V . This situation can be represented in our framework, with the new probability space Ω × V and the probability measure Pε = P ⊗ δε.…”

Section: 2mentioning

confidence: 99%

Linear response for random dynamical systems

Bahsoun

Ruziboev

Saussol

2020

Advances in Mathematics

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We study for the first time linear response for random compositions of maps, chosen independently according to a distribution P. We are interested in the following question: how does an absolutely continuous stationary measure (acsm) of a random system change when P changes smoothly to Pε? For a wide class of one dimensional random maps, we prove differentiability of acsm with respect to ε; moreover, we obtain a linear response formula. We apply our results to iid compositions, with respect to various distributions Pε, of uniformly expanding circle maps, Gauss-Rényi maps (random continued fractions) and Pomeau-Manneville maps. Our results yield an exact formula for the invariant density of random continued fractions; while for Pomeau-Manneville maps our results provide a precise relation between their linear response under certain random perturbations and their linear response under deterministic perturbations.

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“…Results for random systems were proved in [40], 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. Rigorous numerical approaches for the computation of the linear response are available, to some extent, both for deterministic and random systems (see [7,63]).…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2019

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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.

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A rigorous computational approach to linear response

Cited by 29 publications

References 37 publications

Quadratic response of random and deterministic dynamical systems

Quadratic response of random and deterministic dynamical systems

Linear response for random dynamical systems

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

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