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- Title
On the lifting and reconstruction of nonlinear systems with multiple invariant sets.
- Authors
Pan, Shaowu; Duraisamy, Karthik
- Abstract
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.
- Subjects
NONLINEAR systems; LINEAR operators; NONLINEAR operators; DYNAMICAL systems; DISCRETE symmetries; INVARIANT sets
- Publication
Nonlinear Dynamics, 2024, Vol 112, Issue 12, p10157
- ISSN
0924-090X
- Publication type
Article
- DOI
10.1007/s11071-024-09581-0