Topological Dynamics via Learned Hybrid Systems

TL;DR

This paper introduces a method combining switching system identification with topological analysis to better understand the global dynamics of complex hybrid systems from scattered data.

Contribution

It presents a novel integration of convex optimization-based switching system identification with combinatorial topological methods for analyzing hybrid system dynamics.

Findings

Effective computation of Morse graphs from data

Improved topological analysis of hybrid systems

Bridging data-driven methods with topological guarantees

Abstract

The analysis of global dynamics, particularly the identification and characterization of attractors and their regions of attraction, is essential for complex nonlinear and hybrid systems. Combinatorial methods based on Conley's index theory have provided a rigorous framework for this analysis. However, the computation relies on rigorous outer approximations of the dynamics over a discretized state space, which is challenging to obtain from scattered trajectory data. We propose a methodology that integrates recent advances in switching system identification via convex optimization to bridge this gap between data and topological analysis. We leverage the identified switching system to construct combinatorial outer approximations. This paper outlines the integration of these methods and evaluates the efficacy of computing Morse graphs versus data-driven and statistical approaches.