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Description
This dataset contains selected results of rigorous numerical computations conducted in the framework of the research described in the paper "A database schema for the analysis of global dynamics of multiparameter systems" by Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, P. Pilarczyk, published in SIAM Journal on Applied Dynamical Systems (SIADS), Vol. 8, No. 3 (2009), pp. 757–789, doi: 10.1137/080734935.
The parameter space [8,37]✕[3,50] was sampled at the resolution of 50✕50. The phase space was sampled at the resolution of 2048✕2048. A collection of isolating neighborhoods that enclose Morse sets in a Morse decomposition was computed for each box of parameters, and a Conley-Morse graph was determined, with the Conley indices of the Morse sets computed where feasible. Clutching graphs between Morse decompositions found for adjacent boxes were also computed, and the parameter space was subdivided into classes of equivalent Morse decompositions, as described in the paper. The parallelization framework introduced in the paper "Parallelization method for a continuous property" by P. Pilarczyk was used in the computations, as published in Foundations of Computational Mathematics, Vol. 10, No. 1 (2010), 93–114, doi: 10.1007/s10208-009-9050-8.
The dataset contains the Conley-Morse graphs computed for all the parameter boxes. Each graph is encoded in the text format compatible with the "dot" program from the Graphviz Graph Visualization Software package (https://graphviz.org/). All the text files are compressed together in the zipped archive file. The name of each file is in the format "gn_m.txt", where n,m are the integer coordinates of the box, both in the range [0,49]. For example, in order to obtain a PDF file with the visualization of the graph contained in the file "g12_0.txt" one could run the command "dot g12_0.txt -T pdf -o g12_0.pdf". Each node of the graph contains the information on the corresponding isolating neighborhood of a Morse set: the consecutive number of the neighborhood (starting with 0), the number of boxes comprising the neighborhood, and if a Conley index pair was constructed successfully then additionally the homology of the index pair, followed by the index map at each homology level, and the eigenvalues of the map at each homology level. The nodes have different shapes and colors for easy visual identification of the set type; for example, a yellow box indicates an attractor, and a cyan-filled oval stands for an isolating neighborhood with a non-trivial exit set. The directed edges between the nodes show direct connections between the sets (the transitive reduction should be taken for the full set of possible connections).
An interactive browser of all the Conley-Morse graphs and phase space portraits of the Morse decompositions provided in the current series of datasets is available at the address http://www.pawelpilarczyk.com/database/.
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:
-
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CC BY-SAShare-alike - Software:
- GraphViz
Details
- Year of publication:
- 2008
- Verification date:
- 2021-07-29
- Dataset language:
- English
- Fields of science:
-
- mathematics (Natural sciences)
- biological sciences (Natural sciences)
- DOI:
- DOI ID 10.34808/8kpz-bd16 open in new tab
- Series:
- Verified by:
- Gdańsk University of Technology
Keywords
- Dynamical Systems
- Rigorous Numerical Methods
- Conley Index
- Morse Decomposition
- Population Models
- Population Dynamics
- Scientific Computation
References
- publication A Database Schema for the Analysis of Global Dynamics of Multiparameter Systems
- publication Parallelization Method for a Continuous Property
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