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09:39
Spatial Temporal Analysis of Fieldwise Flow in Microvasculature
Published on: November 18, 2019
Cartography of high-dimensional flows: a visual guide to sections and slices
Predrag Cvitanović1, Daniel Borrero-Echeverry, Keith M Carroll
1Center for Nonlinear Science and School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA. predrag@gatech.edu
Chaos (Woodbury, N.Y.)
|January 3, 2013
Summary
This study introduces symmetry reduction using the method of slices to simplify chaotic flows. This technique organizes complex dynamics by mapping solutions onto a reduced state space, aiding in understanding turbulence.
Area of Science:
- Dynamical Systems and Chaos Theory
- Differential Geometry
- Fluid Dynamics
Background:
- Chaotic flows exhibit complex dynamics governed by continuous symmetries.
- Understanding the organization of turbulence and chaos is a significant challenge.
- Traditional methods struggle to analyze high-dimensional state spaces of chaotic systems.
Purpose of the Study:
- To develop a method for simplifying the analysis of chaotic flows.
- To reveal the underlying structure and interrelations of solutions in chaotic systems.
- To visualize and understand the role of invariant solutions in organizing turbulence.
Main Methods:
- Symmetry reduction using the method of slices.
- Quotienting continuous symmetries of chaotic flows.
- Replacing the original state space with a set of charts (atlas).
- Qualitatively capturing solution classes with "templates."
Main Results:
- The method reduces relative equilibria to equilibria and relative periodic orbits to periodic orbits within the slice.
- Visualizations of solutions and their unstable manifolds reveal interrelations.
- The symmetry-reduced slice provides an atlas of dynamically important solution classes.
- The method effectively charts regions of the state space explored by trajectories.
Conclusions:
- Symmetry reduction via the method of slices offers a powerful tool for analyzing chaotic flows.
- This approach simplifies complex dynamics, aiding in the understanding of turbulence organization.
- The atlas of the symmetry-reduced slice provides insights into the structure of chaotic systems.
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