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