Open Access. Powered by Scholars. Published by Universities.®

Applied Mathematics Commons™

Open Access. Powered by Scholars. Published by Universities.®

Mathematics

Chad M. Topaz

Selected Works

Articles 1 - 2 of 2

Full-Text Articles in Applied Mathematics

Topological Data Analysis Of Biological Aggregation Models, Chad M. Topaz, Lori Ziegelmeier, Tom Halverson Apr 2015

Topological Data Analysis Of Biological Aggregation Models, Chad M. Topaz, Lori Ziegelmeier, Tom Halverson

Chad M. Topaz

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 position-velocity 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 …

Go to article

Instabilities And Patterns In Coupled Reaction-Diffusion Layers, Anne J. Catlla, Amelia Mcnamara, Chad M. Topaz Jan 2012

Instabilities And Patterns In Coupled Reaction-Diffusion Layers, Anne J. Catlla, Amelia Mcnamara, Chad M. Topaz

Chad M. Topaz

We study instabilities and pattern formation in reaction-diffusion layers that are diffusively coupled. For two-layer systems of identical two-component reactions, we analyze the stability of homogeneous steady states by exploiting the block symmetric structure of the linear problem. There are eight possible primary bifurcation scenarios, including a Turing-Turing bifurcation that involves two disparate length scales whose ratio may be tuned via the interlayer coupling. For systems of n-component layers and nonidentical layers, the linear problem’s block form allows approximate decomposition into lower-dimensional linear problems if the coupling is sufficiently weak. As an example, we apply these results to a two-layer …

Go to article