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Singularities of two-dimensional invertible piecewise isometric dynamics
1Division of Science and Mathematics, University of Minnesota, Morris, Minnesota 56267, USA. kahng@morris.umn.edu
Chaos (Woodbury, N.Y.)
|July 2, 2009
Summary
This study classifies singularities in 2D invertible piecewise isometric dynamics. One singularity type is removable via manifold lifting, while another drives complex system behavior and initial condition sensitivity.
Area of Science:
- Dynamical Systems Theory
- Geometric Dynamics
- Topology
Background:
- Investigating singularity structure in 2D invertible piecewise isometric dynamics is crucial for understanding system behavior.
- Piecewise isometric dynamics present unique challenges due to their discontinuous nature.
Purpose of the Study:
- To connect the geometric properties of singularities with the overall dynamics of 2D invertible piecewise isometric systems.
- To classify these singularities based on their geometric characteristics.
Main Methods:
- Classification of singularities based on geometric properties.
- Utilizing manifold lifting techniques to analyze singularity removal.
- Analyzing the contribution of different singularity types to system dynamics and orbit structure.
Main Results:
- A classification of 2D bounded invertible piecewise isometric dynamics singularities into three types.
- Demonstration that one singularity type can be eliminated through lifting to a branched manifold.
- Identification of a specific singularity type responsible for complex orbit structures and sensitive dependence on initial conditions.
Conclusions:
- The geometric nature of singularities profoundly impacts the dynamics of 2D invertible piecewise isometric systems.
- Manifold lifting offers a method for simplifying certain singularity types.
- Understanding singularity contributions is key to predicting system behavior, including chaos.
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