🍩 Database of Original & NonTheoretical Uses of Topology
(found 4 matches in 0.002702s)


Topological Data Analysis of Biological Aggregation Models (2015)
Chad M. Topaz, Lori Ziegelmeier, Tom HalversonAbstract
We apply tools from topological data analysis to two mathematical models inspired by biological aggregations such as bird flocks, fish schools, and insect swarms. Our data consists of numerical simulation output from the models of Vicsek and D'Orsogna. These models are dynamical systems describing the movement of agents who interact via alignment, attraction, and/or repulsion. Each simulation time frame is a point cloud in positionvelocity space. We analyze the topological structure of these point clouds, interpreting the persistent homology by calculating the first few Betti numbers. These Betti numbers count connected components, topological circles, and trapped volumes present in the data. To interpret our results, we introduce a visualization that displays Betti numbers over simulation time and topological persistence scale. We compare our topological results to order parameters typically used to quantify the global behavior of aggregations, such as polarization and angular momentum. The topological calculations reveal events and structure not captured by the order parameters. 
A Topological Perspective on Regimes in Dynamical Systems (2021)
Kristian Strommen, Matthew Chantry, Joshua Dorrington, Nina OtterAbstract
The existence and behaviour of socalled `regimes' has been extensively studied in dynamical systems ranging from simple toy models to the atmosphere itself, due to their potential of drastically simplifying complex and chaotic dynamics. Nevertheless, no agreedupon and clearcut definition of a `regime' or a `regime system' exists in the literature. We argue here for a definition which equates the existence of regimes in a system with the existence of nontrivial topological structure. We show, using persistent homology, a tool in topological data analysis, that this definition is both computationally tractable, practically informative, and accounts for a variety of different examples. We further show that alternative, more strict definitions based on clustering and/or temporal persistence criteria fail to account for one or more examples of dynamical systems typically thought of as having regimes. We finally discuss how our methodology can shed light on regime behaviour in the atmosphere, and discuss future prospects. 
Evolutionary Homology on Coupled Dynamical Systems With Applications to Protein Flexibility Analysis (2020)
Zixuan Cang, Elizabeth Munch, GuoWei WeiAbstract
While the spatial topological persistence is naturally constructed from a radiusbased ﬁltration, it has hardly been derived from a temporal ﬁltration. Most topological models are designed for the global topology of a given object as a whole. There is no method reported in the literature for the topology of an individual component in an object to the best of our knowledge. For many problems in science and engineering, the topology of an individual component is important for describing its properties. We propose evolutionary homology (EH) constructed via a time evolutionbased ﬁltration and topological persistence. Our approach couples a set of dynamical systems or chaotic oscillators by the interactions of a physical system, such as a macromolecule. The interactions are approximated by weighted graph Laplacians. Simplices, simplicial complexes, algebraic groups and topological persistence are deﬁned on the coupled trajectories of the chaotic oscillators. The resulting EH gives rise to timedependent topological invariants or evolutionary barcodes for an individual component of the physical system, revealing its topologyfunction relationship. In conjunction with Wasserstein metrics, the proposed EH is applied to protein ﬂexibility analysis, an important problem in computational biophysics. Numerical results for the Bfactor prediction of a benchmark set of 364 proteins indicate that the proposed EH outperforms all the other stateoftheart methods in the ﬁeld.