Linear response for random dynamical systems

Bahsoun, Wael; Ruziboev, Marks; Saussol, Benôıt

doi:10.1016/j.aim.2020.107011

Cited by 32 publications

(38 citation statements)

References 35 publications

(72 reference statements)

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“…The set of Schwartz function is traditionally denoted by S(R). 6 This is a slight abuse of notation.…”

Section: 1mentioning

confidence: 99%

“…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. In the paper [16], like in the present paper in Section 5, general discrete time systems with additive noise are considered, i.e.…”

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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“…The set of Schwartz function is traditionally denoted by S(R). 6 This is a slight abuse of notation.…”

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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“…In the following we will consider two possible choices for the weak norm || || w . One is the usual L 1 norm and the other is the Wasserstein-like norm defined in (8). This norm will be useful when considering perturbations of noise kernels having discontinuities, as the uniformly distributed noise (see Section 4 and in particular Example 38).…”

Section: A General Linear Response Results For Regularizing Markov Opementioning

confidence: 99%

“…Results for this kind of systems were proved in [30], where the technical framework was adapted to stochastic differential equations. Moreover, there is a recent work [8] proving linear response for a class of random uniformly and non uniformly expanding systems. See Remark 16 for a detailed comparison between the present work, and these two closely related results.…”

mentioning

confidence: 99%

A linear response for dynamical systems with additive noise

Galatolo

Giulietti

2019

Nonlinearity

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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.

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“…An example of linear response for small random perturbations of deterministic systems appears in [51]. 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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Linear response for random dynamical systems

Cited by 32 publications

References 35 publications

Quadratic response of random and deterministic dynamical systems

Quadratic response of random and deterministic dynamical systems

A linear response for dynamical systems with additive noise

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

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