Abstract
We discuss recently developed methods for the visualization of dynamical systems which are based on the decomposition of the phase space of the system. The information of the system is represented by a directed graph and then the graph will be decomposed into smaller subsets which eventually defines a partition of the phase space. Depending on the purpose of visualization and the nature of the system, two different decompositions are introduced. The first decomposition algorithm is called Conley-Morse decomposition, which decompose the system according to the gradient-like structure of the system. On the other hand, the latter algorithm, an application of the peer pressure clustering algorithm for directed graphs, decompose each recurrent components of the system into further smaller non-invariant subsets according to the similarity of the dynamical behavior.
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Arai Z, Kalies W, Kokubu H, Mischaikow K, Oka H, Pilarczyk P (2009) A database schema for the analysis of global dynamics of multiparameter systems. SIAM J Appl Dyn Syst 8(3):757–789
Arai Z, Kokubu H, Pilarczyk P (2009) Recent development in rigorous computational methods in dynamical systems. Japan J Ind Appl Math 26(2–3):393–417
Conley C (1978) Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol 38. American Mathematical Society, Providence, RI
Dellnitz M, Junge O (2002) Set oriented numerical methods for dynamical systems? Handbook of dynamical systems, vol 2. North-Holland, pp. 221–264
Kepner J, Gilbert J (eds) (2011) Graph algorithms in the language of linear algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA
Mischaikow K, Mrozek M (2002) Conley index. Handbook of dynamical systems, vol 2. North-Holland, pp 393–460
Szymczak A, Zhang E (2012) Robust Morse decompositions of piecewise constant vector fields. IEEE Trans Vis Comput Graph 18(6):938–951
Szymczak A (2013) Morse connection graphs for piecewise constant vector fields on surfaces. Comput Aided Geom Des 30(6):529–541
The CREST Project Toward a paradigm shift created by mathematics of vortex-boundary interactions (2014). http://www.math.sci.hokudai.ac.jp/crest/
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Arai, Z. (2014). Decomposition and Clustering for the Visualization of Dynamical Systems. In: Anjyo, K. (eds) Mathematical Progress in Expressive Image Synthesis I. Mathematics for Industry, vol 4. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55007-5_3
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DOI: https://doi.org/10.1007/978-4-431-55007-5_3
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