# Spectral Koopman Method for Identifying Stability Boundary

arXiv: https://arxiv.org/abs/2312.06885 · HTML: https://mlsanity.com/paper/2312.06885/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Dec 2023.

## Built on by (most cited first, 5 of 5)

- [Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operator](https://mlsanity.com/paper/2412.16588/) (2024)
- [Learning Koopman-based Stability Certificates for Unknown Nonlinear Systems](https://mlsanity.com/paper/2412.02807/) (2024)
- [Kernel Methods for Stochastic Dynamical Systems with Application to Koopman Eigenfunctions: Feynman-Kac Representations and RKHS Approximation](https://mlsanity.com/paper/2603.01077/) (2026)
- [Kernel Methods for Some Transport Equations with Application to Learning Kernels for the Approximation of Koopman Eigenfunctions: A Unified Approach via Variational Methods, Green's Functions and the Method of Characteristics](https://mlsanity.com/paper/2603.06872/) (2026)
- [Physics-Informed Machine Learning for Characterizing System Stability](https://mlsanity.com/paper/2511.08831/) (2025)

## Abstract (arXiv)

The paper is about characterizing the stability boundary of an autonomous dynamical system using the Koopman spectrum. For a dynamical system with an asymptotically stable equilibrium point, the domain of attraction constitutes a region consisting of all initial conditions attracted to the equilibrium point. The stability boundary is a separatrix region that separates the domain of attraction from the rest of the state space. For a large class of dynamical systems, this stability boundary consists of the union of stable manifolds of all the unstable equilibrium points on the stability boundary. We characterize the stable manifold in terms of the zero-level curve of the Koopman eigenfunction. A path-integral formula is proposed to compute the Koopman eigenfunction for a saddle-type equilibrium point on the stability boundary. The algorithm for identifying stability boundary based on the Koopman eigenfunction is attractive as it does not involve explicit knowledge of system dynamics. We present simulation results to verify the main results of the paper.