# Koopman Resolvent: A Laplace-Domain Analysis of Nonlinear Autonomous Dynamical Systems

arXiv: https://arxiv.org/abs/2009.11544 · HTML: https://mlsanity.com/paper/2009.11544/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Sep 2020.

## Built on by (most cited first, 4 of 4)

- [Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems](https://mlsanity.com/paper/2111.14889/) (2021)
- [Multiplicative Dynamic Mode Decomposition](https://mlsanity.com/paper/2405.05334/) (2024)
- [Learning Koopman-based Stability Certificates for Unknown Nonlinear Systems](https://mlsanity.com/paper/2412.02807/) (2024)
- [Wavelet-Based Observables for Koopman Analysis: An Extended Dynamic Mode Decomposition Framework](https://mlsanity.com/paper/2605.14224/) (2026)

## Abstract (arXiv)

The motivation of our research is to establish a Laplace-domain theory that provides principles and methodology to analyze and synthesize systems with nonlinear dynamics. A semigroup of composition operators defined for nonlinear autonomous dynamical systems -- the Koopman semigroup and its associated Koopman generator -- plays a central role in this study. We introduce the resolvent of the Koopman generator, which we call the Koopman resolvent, and provide its spectral characterization for three types of nonlinear dynamics: ergodic evolution on an attractor, convergence to a stable equilibrium point, and convergence to a (quasi-)stable limit cycle. This shows that the Koopman resolvent provides the Laplace-domain representation of such nonlinear autonomous dynamics. A computational aspect of the Laplace-domain representation is also discussed with emphasis on non-stationary Koopman modes.