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Topological State Space Inference for Dynamical Systems

Mishal Assif P K, Yuliy Baryshnikov

TL;DR

This work tackles nonlinear state-space realization by aiming to recover the topology of the latent state space $M$ from finite, discretely observed trajectories through an output map $g$. It introduces a slack-distance-based bifiltration of trajectory fragments and applies Vietoris–Rips complexes to obtain persistent homology that reflects $M$'s topology. Grounded in Morse theory and topological data analysis, the approach connects the Cohen–Jones–Segal perspective with practical data-driven topology inference, and provides computational methods and complexity insights. Computational demonstrations on the sphere, torus, and Lorenz attractor illustrate that the method recovers expected topological features under realistic sampling and observation constraints, highlighting its potential for topology-guided analysis of dynamical systems.

Abstract

We present a computational pipe aiming at recovery of the topology of the underlying phase space from observation of an output function along a sample of trajectories of a dynamical system.

Topological State Space Inference for Dynamical Systems

TL;DR

This work tackles nonlinear state-space realization by aiming to recover the topology of the latent state space from finite, discretely observed trajectories through an output map . It introduces a slack-distance-based bifiltration of trajectory fragments and applies Vietoris–Rips complexes to obtain persistent homology that reflects 's topology. Grounded in Morse theory and topological data analysis, the approach connects the Cohen–Jones–Segal perspective with practical data-driven topology inference, and provides computational methods and complexity insights. Computational demonstrations on the sphere, torus, and Lorenz attractor illustrate that the method recovers expected topological features under realistic sampling and observation constraints, highlighting its potential for topology-guided analysis of dynamical systems.

Abstract

We present a computational pipe aiming at recovery of the topology of the underlying phase space from observation of an output function along a sample of trajectories of a dynamical system.
Paper Structure (16 sections, 3 theorems, 8 equations, 6 figures, 1 algorithm)

This paper contains 16 sections, 3 theorems, 8 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Setup
  3. Data Analytic setup
  4. Takens' Embedding
  5. Related Results
  6. Mathematical Background
  7. Basic Definitions
  8. Topological Data Analysis
  9. Inferring Topology from Observations
  10. Approximations
  11. Vietoris-Rips Complexes for Slack Distance
  12. Slack Distance and Sliding Windows
  13. Computational Examples
  14. Gradient Dynamical System: Height Function on the 2-Sphere
  15. Gradient Dynamical System: Random Trigonometric Function on the Torus
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $M$ be a compact smooth manifold, and $n \geq 2\dim(M)+1$ an integer. Then, for a generic pair $(\phi, g)$, where $\phi:M\to M$ is a $C^2$-diffeomorphism, and $g$ a function in $C^2(M, \mathbb{R})$, the map is an embedding of $M$.

Figures (6)

  • Figure 1: Segments of trajectories of the gradient of the height function on the sphere. Right: Persistence diagram generated from these trajectories on the sphere. Here $N=400, n=15, t=10, T=1.5$.
  • Figure 2: Segments of trajectories of random vector field on 2-torus. Right: Persistence diagram generated from these segments. Here $N=400, n=25, t=3, T=2.5$
  • Figure 3: Segments of observations of trajectories of random vector field on 2-torus. Right: Persistence diagram generated from these segments. Here $N=650, n=25, t=1, T=2.5$
  • Figure 4: Segments of trajectories of the Lorenz system. Right: Persistence diagram generated from these segments. Here $N=100, n=25, t=20, T=0.25$
  • Figure 5: Segments of observations of trajectories of the Lorenz system. Right: Persistence diagram generated from these segments. Here $N=150, n=25, t=10, T=0.25$
  • ...and 1 more figures

Theorems & Definitions (5)

  • Theorem 1.1
  • Theorem 2.1: Coh95
  • Definition 3.1: Slack distance
  • Lemma 3.2
  • proof