Gene

Quantum mechanics

Algorithm

Persistence (discontinuity)

Engineering

Morse theory

Pure mathematics

Biology

Geotechnical engineering

Dynamics (music)

Cellular homology

Telecommunications

Biochemistry

Computer science

Acoustics

Statistical physics

Genetics

Combinatorial proof

Physics

Homology (biology)

Topological data analysis

Mathematics

Morse homology

Morse code

Simplicial complex

Discrete Morse theory

Combinatorics

Persistent homology

Dynamical systems theory

Chemistry

Topology (electrical circuits)

Algebra over a field

SIAM Journal on Applied Dynamical Systems

We investigate combinatorial dynamical systems on simplicial complexes considered as finite topological spaces. Such systems arise in a natural way from sampling dynamics and may be used to reconstruct some features of the dynamics directly from the sample. We study the homological persistence of Morse decompositions of such systems as a tool for validating the reconstruction. Our approach may be viewed as a step toward applying the classical persistence theory to data collected from a dynamical system. We present experimental results on two numerical examples.

Persistent Homology of Morse Decompositions in Combinatorial Dynamics