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Uniform expansion estimates in the cubic map as a function of the parameter

Opis

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 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 cubic map was analyzed with K=5000, δ=0.01, and a∈[−2,2]. Note that the computations were not mathematically rigorous.

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/.

Plik z danymi badawczymi

a15.dat
97.6 kB, S3 ETag 3511001aca431d2c3e257a51120ba02e-1, pobrań: 57
Hash pliku liczony jest ze wzoru
hexmd5(md5(part1)+md5(part2)+...)-{parts_count} gdzie pojedyncza część pliku jest wielkości 512 MB

Przykładowy skrypt do wyliczenia:
https://github.com/antespi/s3md5

Informacje szczegółowe o pliku

Licencja:
Creative Commons: by-sa 4.0 otwiera się w nowej karcie
CC BY-SA
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Informacje szczegółowe

Rok publikacji:
2008
Data zatwierdzenia:
2021-07-29
Język danych badawczych:
angielski
Dyscypliny:
  • matematyka (Dziedzina nauk ścisłych i przyrodniczych)
DOI:
Identyfikator DOI 10.34808/kb5r-sj53 otwiera się w nowej karcie
Seria:
Weryfikacja:
Politechnika Gdańska

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