Year: 2020
Type: article
Abstract: This work is motivated by the following question in data-driven study of dynamical systems: given a dynamical system that is observed via time series of persistence diagrams that encode topological features of snapshots of solutions, what conclusions can be drawn about solutions of the original dynamical system? We address this challenge in the context of an N dimensional system of ordinary differential equation defined in $${\mathbb {R}}^N$$ R N . To each point in $${\mathbb {R}}^N$$ R N (e.g. an initial condition) we associate a persistence diagram. The main result of this paper is that under this association the preimage of every persistence diagram is contractible. As an application we provide conditions under which multiple time series of persistence diagrams can be used to conclude the existence of a fixed point of the differential equation that generates the time series.
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,750
Awards: G1126465230, INSPIRE: Nonlinear Data Reduction applied to Dense Granular Media, G2515541506, G2682370999, Motivic Cohomology, Motivic Homotopy Theory and K-theory, Collaborative Research: Computational and Data-Enabled Science and Engineering: Characterizing Dynamics of Particle-based Systems, G4636648896, G4713059963, G5270150132, CDI-TYPE II--COLLABORATIVE RESEARCH: Using Algebraic Topology to Connect Models with Measurements in Complex Nonequilibrium Systems, G6182062231, G668241585, Needle in a haystack: effective search for dynamics across the space of networks, Motivic Cohomology, Motivic Homotopy Theory, and K-Theory, G862860306, G8818612554
OpenAlex ID: W2899460079