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

Yoshihiko Susuki, Alexandre Mauroy, Igor Mezic
2021 arXiv   pre-print
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
more » ... e 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.
arXiv:2009.11544v2 fatcat:cc5nljq46zds7lwhuu5s6ppmvy