On the lifting and reconstruction of nonlinear systems with multiple invariant sets
This work clarifies misconceptions in dynamical systems theory, offering incremental improvements for researchers in control or physics dealing with complex nonlinear systems.
The paper addresses the challenge of applying Koopman operators to nonlinear dynamical systems with multiple disjoint invariant sets, such as basins of attraction, by explaining the linear reconstruction mechanism and using discrete symmetry to construct eigenfunctions more efficiently, with numerical examples demonstrating improved learning.
The Koopman operator provides a linear perspective on non-linear dynamics by focusing on the evolution of observables in an invariant subspace. Observables of interest are typically linearly reconstructed from the Koopman eigenfunctions. Despite the broad use of Koopman operators over the past few years, there exist some misconceptions about the applicability of Koopman operators to dynamical systems with more than one disjoint invariant sets (e.g., basins of attractions from isolated fixed points). In this work, we first provide a simple explanation for the mechanism of linear reconstruction-based Koopman operators of nonlinear systems with multiple disjoint invariant sets. Next, we discuss the use of discrete symmetry among such invariant sets to construct Koopman eigenfunctions in a data efficient manner. Finally, several numerical examples are provided to illustrate the benefits of exploiting symmetry for learning the Koopman operator.