Discrete bifurcation diagrams and persistence

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