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Cutting and shuffling a hemisphere: Nonorthogonal axes.
Thomas F Lynn1, Lachlan D Smith2, Julio M Ottino3
1Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA.
Physical Review. E
|April 20, 2019
Summary
This study explores cutting-and-shuffling dynamics of a hemispherical shell using piecewise isometry (PWI) theory. Nonorthogonal rotation axes enhance mixing and reveal new periodic dynamics with polygonal tilings.
Area of Science:
- Geometric dynamics
- Chaos theory
- Computational physics
Background:
- Piecewise isometry (PWI) theory models systems with cutting and shuffling.
- Previous studies on hemispherical shell dynamics were limited by orthogonal rotation axes.
Purpose of the Study:
- To investigate the effects of nonorthogonal rotation axes on the cutting-and-shuffling dynamics of a hemispherical shell.
- To develop and utilize a new computational method for efficient parameter sweeps and analysis.
- To explore novel mixing protocols and their resulting dynamics.
Main Methods:
- Application of piecewise isometry (PWI) theory.
- Relaxation of restrictions on rotation axes to include nonorthogonal configurations.
- Development of a parallel processing computational method for cutting-and-shuffling simulations.
- Extensive parameter sweeps to analyze mixing protocols.
Main Results:
- Nonorthogonal rotation axes increase the average degree of mixing compared to orthogonal axes.
- Arnold tongues and their intersections still lead to poor mixing in specific parameter regions.
- A new class of periodic protocols with polygonal tilings was discovered due to Arnold tongue intersections.
Conclusions:
- Nonorthogonal rotation axes offer greater flexibility and improved mixing in hemispherical shell dynamics.
- The study identifies specific conditions leading to periodic dynamics and highlights the role of symmetry breaking.
- The developed computational method enables more comprehensive investigations into complex dynamical systems.
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