Stochastic intervals for the family of quadratic maps

Description

Numerical analysis of chaotic dynamics is a challenging task. The one-parameter families of logistic maps and closely related quadratic maps f_a(x)=a-x^2 are well-known examples of such dynamical systems. Determining parameter values that yield stochastic-like dynamics is especially difficult, because although this set has positive Lebesgue measure, it turns out to be nowhere dense. One of the steps in computing a rigorous lower bound on this measure is determining intervals of parameters such that the iterates of the critical point by the sets of corresponding maps become large. The dataset is a result of such a computation, described in details in the paper "Rigorous numerics for the quadratic family" by A. Golmakani, C. E. Koudjinan, S. Luzzatto, and P. Pilarczyk, arXiv:2004.13444v1 [math.DS] (May 2020) URL of the paper: https://arxiv.org/abs/2004.13444v1

Algorithms developed in the paper were used to compute rigorous bounds for the orbits of the critical points for intervals of parameter values in the quadratic family of maps. A dynamically defined partition P of the parameter interval Ω=[1.4,2] into almost 4 million subintervals was computed, and data for 1,436,063 intervals that yielded large images was gathered. These intervals are suspected of containing a positive measure of parameters for which the quadratic map exhibits chaotic dynamics in the ergodic sense, and are thus called "stochastic" intervals. In addition to the intervals themselves and their images, rigorous bounds on certain derivatives was also computed and is provided in the dataset.

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intervals.zip

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Year of publication:

2020

Verification date:

2020-12-17

Dataset language:

English

Fields of science:

mathematics (Natural sciences)

DOI:

10.34808/szfn-gv40

Verified by:

Gdańsk University of Technology

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