Koopman principle eigenfunctions and linearization of diffeomorphisms
Ryan Mohr, Igor Mezić
TL;DR
The paper analyzes a nonlinear diffeomorphism $T$ on a finite-dimensional complex Banach space with an asymptotically stable hyperbolic fixed point, linking the principle eigenfunctions of the Koopman operator $U_T$ to the existence of a local topological conjugacy with the linearization $T'(0)$. By constructing a principle algebra $\,\mathcal{A}_{T'(0)}$ from non-resonant eigenfunctions and generating approximate conjugacies via a normal-form-like procedure, the authors show how to transfer eigenfunction expansions to the nonlinear setting and, under suitable convergence, obtain a genuine conjugacy whose pullback preserves spectral structure. They prove that, in real spaces, the uniform closures of the principle and pullback algebras are dense in the space of continuous observables (or those vanishing at the fixed point) by Stone–Weierstrass, implying any continuous observable can be approximated by observables with Koopman spectral expansions. Overall, the work provides a rigorous framework for choosing and approximating observables for nonlinear dynamics with complete Koopman spectral representations near a fixed point, via a sequence of polynomial conjugacies derived from principle eigenfunctions.
Abstract
This paper considers a nonlinear dynamical system on a complex, finite dimensional Banach space which has an asymptotically stable, hyperbolic fixed point. We investigate the connection between the so-called principle eigenfunctions of the Koopman operator and the existence of a topological conjugacy between the nonlinear dynamics and its linearization in the neighborhood of the fixed point. The principle eigenfunctions generate an algebra of observables for the linear dynamics --- called the principle algebra --- which can be used to generate a sequence of approximate conjugacy maps in the same manner as it is done in normal form theory. Each element of the principle algebra has an expansion into eigenfunctions of the Koopman operator and composing an eigenfunction with one of the approximate topological conjugacies gives an approximate eigenfunction of the Koopman operator associated with the nonlinear dynamical system. When the limit of the approximate conjugacies exists and attention is restricted to real Banach spaces, a simple application of the Stone-Weierstrass theorem shows that both the principle algebra and the pull-back algebra --- defined by composing the principle algebra with the topological conjugacy --- are uniformly dense in either the space of continuous functions or the maximal ideal of continuous functions vanishing at the fixed point. The point is that, a priori, it is difficult to know which space of observables to use for dissipative nonlinear dynamical systems whose elements have spectral expansions into eigenfunctions. These results say that any continuous observable is arbitrarily close to one that has such an expansion.