Uniform expansion estimates in the quadratic map with the smallest critical neighborhood for which the expansion exponent λ is positive - Open Research Data - Bridge of Knowledge

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Uniform expansion estimates in the quadratic map with the smallest critical neighborhood for which the expansion exponent λ is positive

Description

This dataset contains selected results of numerical computations described in the paper "Quantitative hyperbolicity estimates in one-dimensional dynamics" by S. Day, H. Kokubu, S. Luzzatto, K. Mischaikow, H. Oka, P. Pilarczyk, published in Nonlinearity, Vol. 21, No. 9 (2008), 1967-1987, doi: 10.1088/0951-7715/21/9/002.

Using an approach based on graph algorithms described in the paper, comprehensive computation was conducted to obtain a rigorous lower bound on the expansion exponent λ (Statement 1 in the paper), the constant C (Statement 1 in the paper), and the adjusted exponent λ0 (Statement 2 in the paper). The size of the partition into which the domain of the map was subdivided is denoted by K. The radius of the critical neighborhood is denoted by δ. The key parameter of the map is denoted by a. The parallelization framework introduced in the paper "Parallelization method for a continuous property" by P. Pilarczyk, as published in Foundations of Computational Mathematics, Vol. 10, No. 1 (2010), 93–114, doi: 10.1007/s10208-009-9050-8, was used in order to use several CPUs at a time. Note that the data records are not sorted.

In this specific computation, the quadratic map was analyzed with K=5000, a∈[1.7,2.0], and δ chosen to be as small as possible, assuming that still λ>0 can be obtained. This dataset was a basis for Figures 7 and 9 in the paper.

The data is provided in plain text format. Each line with comments begins with a semicolon, each line with a single data record begins with an asterisk. Each data record consists of the following items, separated with the space:

  • the asterisk that indicates the beginning of a data record
  • the identifier of the data record in the format level:number (e.g., 4:12)
  • the left endpoint of the parameter interval (minimal parameter value)
  • the right endpoint of the parameter interval (maximal parameter value)
  • the total number of intervals that cover both the critical neighborhood and the remainder of the domain of the map
  • the diameter δ of the critical neighborhood
  • the computed expansion exponent λ
  • the computed value of log C (0 if not computed)
  • the computed value of λ0 (0 if not computed)
  • the computation time (in seconds)

The actual software that was used to obtain the results and also some illustrations are available at http://www.pawelpilarczyk.com/unifexp/.

Dataset file

a10.dat
403.1 kB, S3 ETag 2b570a56f86bcf8eb5f76e6ec9d52a36-1, downloads: 66
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hexmd5(md5(part1)+md5(part2)+...)-{parts_count} where a single part of the file is 512 MB in size.

Example script for calculation:
https://github.com/antespi/s3md5

File details

License:
Creative Commons: by-sa 4.0 open in new tab
CC BY-SA
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Details

Year of publication:
2008
Verification date:
2021-07-29
Dataset language:
English
Fields of science:
  • mathematics (Natural sciences)
DOI:
DOI ID 10.34808/x49e-gq14 open in new tab
Series:
Verified by:
Gdańsk University of Technology

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