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 trajectorypersistent homology groups approximating local phase flow topology (to some resolution)

Approach

Step 1: Poincaré Sections surface of section

Step 2: Metric Space Embedding

Step 3: Simplicial Complex Filtration R 0 

Step 4: Persistent Homology Calculation R 0  R 0 1 dim

Results: Example 1

Results: Example 2

Results: Example 3

Acknowledgements Dr. Martin Lo NASA Jet Propulsion Laboratory Prof. Olivier de Weck Massachusetts Institute of Technology Henry Adams Stanford 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