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
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
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
Problem Description
numerically integrated trajectory persistent homology groups approximatinglocal phase flow topology (to some resolution)
Approach
Step 1: Poincaré Sections
surface of section
Step 2: Metric Space Embedding
Step 2: Metric Space Embedding
Step 3: Simplicial Complex Filtration
R0
Step 4: Persistent Homology Calculation
R0 R
0
1
dim
Results: Example 1
Results: Example 2
Results: Example 3
Acknowledgements
Dr. Martin LoNASA Jet Propulsion Laboratory
Prof. Olivier de WeckMassachusetts Institute of Technology
Henry AdamsStanford University
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
Reference
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
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
Earth-Moon Hill’s Regions
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