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Topological Dynamics via Learned Hybrid Systems

Bernardo Rivas, Kaito Iwasaki, William Kalies, Anthony Bloch, Maani Ghaffari

TL;DR

Problem: robustly extracting global dynamical structure from trajectory data for nonlinear and hybrid systems using Conley index theory. The main approach identifies switching vector fields via convex optimization, producing an interpretable model $\hat{f}(x)=\sum_{j=1}^M \hat{\lambda}_j(x)\hat{f}_j(x)$ that yields a provable outer-approximation $\mathcal{F}$ of the time-$\tau$ map. The paper develops a bilevel SDP/LP scheme for mode assignment and dynamics estimation, then uses the resulting combinatorial dynamics to compute Morse graphs and Regions of Attraction via $\mathsf{MG}(\mathcal{F})$ and $RoA$. Experiments on Toggle Switch and Piecewise Van der Pol show improved fidelity of the recovered topological structure relative to Gaussian Process and Lipschitz baselines, with interpretable switching surfaces and invariants.

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.

Topological Dynamics via Learned Hybrid Systems

TL;DR

Problem: robustly extracting global dynamical structure from trajectory data for nonlinear and hybrid systems using Conley index theory. The main approach identifies switching vector fields via convex optimization, producing an interpretable model that yields a provable outer-approximation of the time- map. The paper develops a bilevel SDP/LP scheme for mode assignment and dynamics estimation, then uses the resulting combinatorial dynamics to compute Morse graphs and Regions of Attraction via and . Experiments on Toggle Switch and Piecewise Van der Pol show improved fidelity of the recovered topological structure relative to Gaussian Process and Lipschitz baselines, with interpretable switching surfaces and invariants.

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.

Paper Structure

This paper contains 10 sections, 1 theorem, 15 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Topological Dynamics
  3. Switching System Identification
  4. Combinatorial Dynamics
  5. Outer Approximation
  6. Morse Graph and Regions of Attraction
  7. Examples
  8. Toggle Switch: Bistability in Gene Regulation
  9. Piecewise Van der Pol: Limit Cycle
  10. Conclusion

Key Result

Proposition 4.4

If $\|\hat{\varphi}_\tau - \varphi_\tau\| \leq \delta$ for some $0<\delta< \mathop{\mathrm{diam}}\nolimits(\mathcal{X})/2$, then the coarser outer approximation of $\hat{\varphi}_\tau$ is an outer approximation of $\varphi_\tau$. Furthermore, if $\delta < d(\varphi_\tau(|\xi|), \partial|\mathcal{F}(\xi)|)$ where then $\mathcal{F}_c = \mathcal{F}$.

Figures (6)

  • Figure 1: Outer approximation of time-$\tau$ map via the bounding box $\mathcal{R}_\tau(|\xi|)$.
  • Figure 2: Learned switching linear system colored by learned modes.
  • Figure 3: Morse sets (top) and Morse graphs (bottom).
  • Figure 4: Aggregated Morse graph (a) and geometric realization of Morse sets and regions of attraction.
  • Figure 5: Learned switching polynomial system colored by learned modes.
  • ...and 1 more figures

Theorems & Definitions (7)

  • remark 1
  • Definition 4.1
  • Remark 4.2
  • Definition 4.3
  • Proposition 4.4
  • proof
  • Definition 4.5