Uniform expansion estimates in the quadratic map with the smallest critical neighborhood for which the expansion exponent λ0 is greater than 0.1
Description
This dataset contains selected results of numerical computations described in the paper "Quantitative hyperbolicity estimates in onedimensional dynamics" by S. Day, H. Kokubu, S. Luzzatto, K. Mischaikow, H. Oka, P. Pilarczyk, published in Nonlinearity, Vol. 21, No. 9 (2008), 19671987, doi: 10.1088/09517715/21/9/002.
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/s1020800990508, 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 λ0>0.1 can be obtained.
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
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:

open in new tabCC BYSASharealike
Details
 Year of publication:
 2008
 Verification date:
 20210729
 Dataset language:
 English
 Fields of science:

 mathematics (Natural sciences)
 DOI:
 DOI ID 10.34808/09aekh19 open in new tab
 Series:
 Verified by:
 Gdańsk University of Technology
Keywords
 Dynamical Systems
 Rigorous Numerical Methods
 Scientific Computation
 OneDimensional Dynamics
 Unimodal Map
 Quadratic Map
 Lyapunov Exponent
References
 publication Quantitative hyperbolicity estimates in onedimensional dynamics
 publication Parallelization Method for a Continuous Property
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