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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.

Dataset file

intervals.zip
86.7 MB, MD5 04c833bd6f726b7354b657e03a4467db-1, downloads: 0
download file intervals.zip

File details

License:
CC BY
Attribution

Details

Year of publication:
2020
Dataset language:
English
Fields of science:
  • Mathematics (Natural sciences)
DOI:
DOI ID 10.34808/szfn-gv40 open in new tab
Verified by:
Gdańsk University of Technology

Keywords

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