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Persistent Homology of Morse Decompositions in Combinatorial Dynamics
We investigate combinatorial dynamical systems on simplicial complexes c...
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Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems
Bifurcations in dynamical systems characterize qualitative changes in th...
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Filtration Simplification for Persistent Homology via Edge Contraction
Persistent homology is a popular data analysis technique that is used to...
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Persistence-perfect discrete gradient vector fields and multi-parameter persistence
The main objective of this paper is to introduce and study a notion of p...
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Persistence in sampled dynamical systems faster
We call a continuous self-map that reveals itself through a discrete set...
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Finding and Classifying Critical Points of 2D Vector Fields: A Cell-Oriented Approach Using Group Theory
We present a novel approach to finding critical points in cell-wise bary...
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Assessing monotonicity of transfer functions in nonlinear dynamical control systems
When dealing with dynamical systems arising in diverse control systems, ...
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Persistence of the Conley Index in Combinatorial Dynamical Systems
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.
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