Discrete bifurcation diagrams and persistence

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2018

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Örnek, Türkmen

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Let fti : M → R be a discrete Morse function on a cell complex M for each t0 < t1 < ... < tn = 1. Let us denote slice as Mi = M ×{ti} ⊂ M × I and let Vi be the discrete vector ﬁeld on each slice. After extending the discrete vector ﬁeld on each slice to a discrete vector ﬁeld on all of M ×I, a discrete bifurcation diagram is obtained by connecting critical cells of the slices. In”Birth and Death in Discrete MorseTheory”(King,Knudson,Mramor), a solution about ﬁnding the discrete bifurcation diagram has been presented. In this article, two cases have been researched. In the ﬁrst case, triangulation of each slice is the same andM×I is regularly celldecomposed. For the ﬁrst case,there is a known algorithm about extending discrete vector ﬁeld on each slice to a gradient vector ﬁeld on all of M×I. In the second case, triangulation on each slice of cell complex M is different. It has seen that the way to extend the vector ﬁeld on each slice to the vector ﬁeld on all of M ×I is not obvious. In this case, we have to choose a cell structure of M ×I compatible with slices. The algorithm how one could deﬁne a discrete vector ﬁeld on the resulting cell complex which does not have any closed paths is not known. In this thesis, we intend to explain how to deﬁne the extended vector ﬁeld when the triangulation on M is different and obtain the discrete bifurcation diagram. Furthermore, we want to get discrete bifurcation diagrams by using the persistence diagrams and compare these two discrete bifurcation diagrams.

Subject Keywords

Morse theory., Homology theory., Bifurcation theory.

URI

http://etd.lib.metu.edu.tr/upload/12622428/index.pdf
https://hdl.handle.net/11511/27745

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Graduate School of Natural and Applied Sciences, Thesis

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Citation Formats

T. Örnek, “Discrete bifurcation diagrams and persistence,” Ph.D. - Doctoral Program, Middle East Technical University, 2018.