1,720,966 research outputs found
QUANTITATIVE STATISTICAL STABILITY AND LINEAR RESPONSE FOR IRRATIONAL ROTATIONS AND DIFFEOMORPHISMS OF THE CIRCLE
No full textWe prove quantitative statistical stability results for a large class of small C0 perturbations of circle diffeomorphisms with irrational rotation numbers. We show that if the rotation number is Diophantine the invariant measure varies in a Hölder way under perturbation of the map and the Hölder exponent depends on the Diophantine type of the rotation number. The set of admissible perturbations includes the ones coming from spatial discretization and hence numerical truncation. We also show linear response for smooth perturbations that preserve the rotation number, as well as for more general ones. This is done by means of classical tools from KAM theory, while the quantitative stability results are obtained by transfer operator techniques applied to suitable spaces of measures with a weak topology
Spectral gap and quantitative statistical stability for systems with contracting fibers and lorenz-like maps
No full textWe consider transformations preserving a contracting foliation, such that the associated quotient map satisfies a Lasota-Yorke inequality. We prove that the associated transfer operator, acting on suitable normed spaces, has a spectral gap (on which we have quantitative estimation). As an application we consider Lorenz-like two dimensional maps (piecewise hyperbolic with unbounded contraction and expansion rate): we prove that those systems have a spectral gap and we show a quantitative estimate for their statistical stability. Under deterministic perturbations of the system of size δ, the physical measure varies continuously, with a modulus of continuity O(δ log δ), which is asymptotically optimal for this kind of piecewise smooth maps
Quadratic response of random and deterministic dynamical systems
No full textWe 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
Existence of noise induced order, a computer aided proof
No full textWe prove the existence of noise induced order in the Matsumoto-Tsuda model, where it was originally discovered in 1983 by numerical simulations. This is a model of the famous Belousov-Zhabotinsky reaction, a chaotic chemical reaction, and consists of a one dimensional random dynamical system with additive noise. The simulations showed that an increase in amplitude of the noise causes the Lyapunov exponent to decrease from positive to negative; we give a mathematical proof of the existence of this transition. The method we use relies on some computer aided estimates providing a certified approximation of the system's stationary measure in the L 1 norm. This is realized by explicit functional analytic estimates working together with an efficient algorithm. The method is general enough to be adapted to any piecewise differentiable dynamical system on the unit interval with additive noise. We also prove that the stationary measure varies in a Lipschitz way if the system is perturbed and that the Lyapunov exponent of the system varies in a Hölder way when the noise amplitude increases
A general framework for the rigorous computation of invariant densities and the coarse-fine strategy
Full text linkIn this paper we present a general, axiomatical framework for the rigorous approximation of invariant densities and other important statistical features of dynamics. We approximate the system through a finite element reduction, by composing the associated transfer operator with a suitable finite dimensional projection (a discretization scheme) as in the well-known Ulam method.
We introduce a general framework based on a list of properties (of the system and of the projection) that need to be verified so that we can take advantage of a so-called “coarse-fine” strategy. This strategy is a novel method in which we exploit information coming from a coarser approximation of the system to get useful information on a finer approximation, speeding up the computation. This coarse-fine strategy allows a precise estimation of invariant densities and also allows to estimate rigorously the speed of mixing of the system by the speed of mixing of a coarse approximation of it, which can easily be estimated by the computer.
The estimates obtained her e are rigorous, i.e., they come with exact error bounds that are guaranteed to hold and take into account both the discretization and the approximations induced by finite-precision arithmetic.
We apply this framework to several discretization schemes and examples of invariant density computation from previous works, obtaining a remarkable reduction in computation time.
We have implemented the numerical methods described here in the Julia programming language, and released our implementation publicly as a Julia package
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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