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

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.

24 pages, 2 figures

References in corpus (4)

• Applied Koopmanism
• Ergodic theory, Dynamic Mode Decomposition and Computation of Spectral Properties of the Koopman operator
• On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator
• Construction of eigenfunctions for scalar-type operators via Laplace averages with connections to the Koopman operator