Algorithmic Classification of Resonant Orbits Using Persistent Homology in Poincaré Sections Thomas...
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Algorithmic Classification of Resonant Orbits Using Persistent Homology in Poincaré Sections
Thomas Coffee
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Motivation• Resonance structures offer powerful insight into
global phase flow in nonintegrable dynamical systems such as the restricted 3-body problem
• (Quasi)periodic orbits are themselves frequently important for practical mission design
• Classical methods for analyzing resonance structures (perturbations, visual methods) pose challenges in high-dimensional phase spaces
• Desired: a targetable, scalable algorithmic approach for analysis of resonance structures in arbitrary dimensions
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Outline
• Problem Description
• Approach– Step 1: Poincaré Sections– Step 2: Metric Space Embedding– Step 3: Simplicial Complex Filtration– Step 4: Persistent Homology Calculation
• Results
• Contributions
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Problem Description
numerically integrated trajectory persistent homology groups approximatinglocal phase flow topology (to some resolution)
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Approach
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Step 1: Poincaré Sections
surface of section
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Step 2: Metric Space Embedding
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Step 2: Metric Space Embedding
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Step 3: Simplicial Complex Filtration
R0
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Step 4: Persistent Homology Calculation
R0 R
0
1
dim
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Results: Example 1
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Results: Example 2
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Results: Example 3
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Acknowledgements
Dr. Martin LoNASA Jet Propulsion Laboratory
Prof. Olivier de WeckMassachusetts Institute of Technology
Henry AdamsStanford University
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Contributions
• Developed method to compute multiscale metric in finite Poincaré sections reflecting underlying topology
• Demonstrated scalable numeric approach for identifying targeted resonance structures in arbitrary computable dynamical systems of any dimension
• Implemented and applied this approach to simple examples in the planar and spatial circular restricted three-body problem
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Reference
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Background: Persistent Homology
• Edelsbrunner et al. (2002) developed an efficient algorithm to generate persistent homology groups from point clouds embedded in a metric space
• Zomorodian & Carlsson (2003) generalized this algorithm to arbitrary dimensions
• de Silva & Carlsson (2004) introduced an efficient approximation algorithm using a set of landmark points selected from point data
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Background: Nonlinear Dimensionality Reduction
• Tenenbaum et al. (2000) used shortest paths in sparse weighted graph to construct global metric from local geometry
• de Silva & Tenenbaum (2003) scaled edge weights by local density to learn conformal maps with underlying uniform sampling
• Yang (2006) used local linear model fitting and neighborhood size selection to reduce distortion of locally linear embeddings
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Earth-Moon Hill’s Regions
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