Persistence of the Conley Index in Combinatorial Dynamical Systems

03/12/2020
by   Tamal K. Dey, et al.
0

A combinatorial framework for dynamical systems provides an avenue for connecting classical dynamics with data-oriented, algorithmic methods. Combinatorial vector fields introduced by Forman and their recent generalization to multivector fields have provided a starting point for building such a connection. In this work, we strengthen this relationship by placing the Conley index in the persistent homology setting. Conley indices are homological features associated with so-called isolated invariant sets, so a change in the Conley index is a response to perturbation in an underlying multivector field. We show how one can use zigzag persistence to summarize changes to the Conley index, and we develop techniques to capture such changes in the presence of noise. We conclude by developing an algorithm to track features in a changing multivector field.

READ FULL TEXT

page 10

page 11

page 12

page 14

page 21

research
07/05/2021

Persistence of Conley-Morse Graphs in Combinatorial Dynamical Systems

Multivector fields provide an avenue for studying continuous dynamical s...
research
03/11/2022

Tracking Dynamical Features via Continuation and Persistence

Multivector fields and combinatorial dynamical systems have recently bec...
research
01/19/2018

Persistent Homology of Morse Decompositions in Combinatorial Dynamics

We investigate combinatorial dynamical systems on simplicial complexes c...
research
05/06/2021

Floer Homology: From Generalized Morse-Smale Dynamical Systems to Forman's Combinatorial Vector Fields

We construct a Floer type boundary operator for generalised Morse-Smale ...
research
09/18/2020

Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems

Bifurcations in dynamical systems characterize qualitative changes in th...
research
03/05/2023

Computing Connection Matrices via Persistence-like Reductions

Connection matrices are a generalization of Morse boundary operators fro...
research
09/12/2017

Persistence in sampled dynamical systems faster

We call a continuous self-map that reveals itself through a discrete set...

Please sign up or login with your details

Forgot password? Click here to reset