- Project Year:
2021
- REU Student (s):
Anna Cusenza | University of California-Los Angeles CA
- Student 1 Institution:
University of California-Los Angeles
- Project Mentor:
Konstantin Mischaikow
- Project Mentor Area:
Mathematics
- Project Abstract:
Our goal is to interpret the dynamics of multi-parameter nonlinear dynamical systems. Such systems can have very complicated behavior. Because of this and the fact that in applications data collected may not accurately represent the generating system, it is desired to gather from the data robust patterns which are consistent with respect to perturbations in parameter values. A method to analyze data generated by such systems, as introduced in previous work, is to discretize the dynamics via a decomposition of the compact phase space into square grids. With a grid decomposition of the phase space, a directed graph which approximates the behavior of the system for a fixed set of parameter values is created. Each node in the directed graph represents a cubical cell in the phase space, and the directed edges denote where each cell maps to under the system. With graph algorithms we can find regions which contain recurrent dynamics, and using Conley index theory we can further identify special properties of the system. In this project we use the method of analysis described above, with the change being that we decompose the phase space using Voronoi cells rather than cubical cells. A Voronoi cell complex is generated by a finite set $P = \{x_1,x_2,\dots, x_n\}$, where each $x_i$ is chosen in the phase space. Each Voronoi cell $V_i$ is defined by $V_i = \{x|d(x,x_i)\leq d(x,x_j) \text{ for all } j\neq i,\ 1\leq j\leq n\}$. Our goal is to determine whether using Voronoi cells to decompose the phase space yields comparable results to the results generated by the cubical complex; in visual quality and accuracy. The goal is to apply these methods to real data collected from systems generated by weather patterns and robotics.
