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

TL;DR

This paper introduces the Koopman resolvent, a Laplace-domain tool for analyzing nonlinear autonomous systems, providing spectral characterization for various dynamics and facilitating system analysis and synthesis.

Contribution

It develops the Koopman resolvent and characterizes its spectral properties for different nonlinear dynamics types, advancing Laplace-domain analysis of such systems.

Findings

Spectral characterization of Koopman resolvent for ergodic dynamics

Analysis of convergence to equilibrium and limit cycles

Discussion on computational aspects and non-stationary modes

Abstract

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.