[Paper Review] Generic Properties of Koopman Eigenfunctions for Stable Fixed Points and Periodic Orbits

Matthew D. Kvalheim, Hong, David+1 more|arXiv (Cornell University)|Oct 7, 2020

Model Reduction and Neural NetworksPhysics and Astronomy86 references57 citations

TL;DR

This paper establishes that for 'typical' smooth (C∞) vector fields with attracting hyperbolic fixed points or periodic orbits, the C∞ Koopman eigenfunctions can be completely classified. Building on prior existence and uniqueness results for Koopman eigenfunctions, the authors prove that under generic conditions—specifically ∞-nonresonance of the linearized dynamics—these eigenfunctions are finite linear combinations of products of principal eigenfunctions and their complex conjugates, ensuring well-posedness and global structure for applications in nonlinear dynamics and model reduction.

ABSTRACT

Our recent work established existence and uniqueness results for $\mathcal{C}^{k,\alpha}_{ ext{loc}}$ globally defined linearizing semiconjugacies for $\mathcal{C}^1$ flows having a globally attracting hyperbolic fixed point or periodic orbit (Kvalheim and Revzen, 2019). Applications include (i) improvements, such as uniqueness statements, for the Sternberg linearization and Floquet normal form theorems; (ii) results concerning the existence, uniqueness, classification, and convergence of various quantities appearing in the "applied Koopmanism" literature, such as principal eigenfunctions, isostables, and Laplace averages. In this work we give an exposition of some of these results, with an emphasis on the Koopmanism applications, and consider their broadness of applicability. In particular we show that, for "almost all" $\mathcal{C}^\infty$ flows having a globally attracting hyperbolic fixed point or periodic orbit, the $\mathcal{C}^\infty$ Koopman eigenfunctions can be completely classified, generalizing a result known for analytic systems. For such systems, every $\mathcal{C}^\infty$ eigenfunction is uniquely determined by its eigenvalue modulo scalar multiplication.

Motivation & Objective

• To establish the broad applicability of Koopman eigenfunction classification results for nonlinear dynamical systems with stable attractors.
• To address the long-standing problem of uniqueness and existence for Koopman eigenfunctions, particularly in the context of isostables and Laplace averages.
• To show that the strong classification results for C∞ Koopman eigenfunctions in prior work (Kvalheim and Revzen, 2019) hold not just in special cases but for 'typical' C∞ vector fields.
• To provide a theoretical foundation for applied Koopmanism by proving that key quantities like isostables and isostable coordinates are well-posed and uniquely determined under generic conditions.
• To extend the known classification of analytic eigenfunctions to the smooth (C∞) setting using genericity arguments in infinite-dimensional function spaces.

Proposed method

• Uses differential topology and genericity theory in the space of C∞ vector fields to show that the nonresonance condition (a key hypothesis in prior work) holds on open and dense subsets.
• Applies the concept of ∞-nonresonance between the linearized flow at the fixed point or periodic orbit and its own adjoint, ensuring spectral separation.
• Employs a limiting procedure (via time-averaged observables) to construct Koopman eigenfunctions from approximate eigenfunctions, leveraging convergence in the Ck compact-open topology.
• Utilizes the theory of linearizing semiconjugacies to reduce the problem of eigenfunction classification to the spectral properties of the linearized system.
• Applies results from symmetric polynomials and eigenvalue perturbation theory to analyze the structure of eigenfunctions under generic conditions.
• Combines theorems from dynamical systems (e.g., Hartman-Grobman, Sternberg) with functional analytic techniques to prove existence and uniqueness of eigenfunctions in C∞ regularity.

Experimental results

Research questions

• RQ1 Under what generic conditions do C∞ Koopman eigenfunctions for stable fixed points or periodic orbits exist and admit a complete classification?
• RQ2 How broad is the applicability of the classification result for C∞ Koopman eigenfunctions beyond analytic systems?
• RQ3 Is the uniqueness of principal eigenfunctions robust under small perturbations of the vector field?
• RQ4 Can the existence and uniqueness of isostables and isostable coordinates be guaranteed generically in the C∞ setting?
• RQ5 What is the precise role of ∞-nonresonance in ensuring the completeness and uniqueness of the eigenfunction classification?

Key findings

• For a generic C∞ vector field with an attracting hyperbolic fixed point, the C∞ Koopman eigenfunctions are completely classified as finite linear combinations of products of principal eigenfunctions and their complex conjugates.
• The classification holds generically—i.e., for an open and dense set of C∞ vector fields—under the condition of ∞-nonresonance between the linearized system and its adjoint.
• The principal eigenfunctions are uniquely determined by their initial derivative at the fixed point, and can be constructed via a limiting procedure involving time-averaged observables.
• For periodic orbits, the eigenfunctions include a phase eigenfunction ψθ with eigenvalue i2π/τ, and all other eigenfunctions are finite combinations of products of ψθ with eigenfunctions of the stable subspace.
• The results generalize the known classification for analytic systems to the smooth (C∞) case, showing that the structural properties of Koopman eigenfunctions are robust under generic perturbations.
• The spectral spread condition in prior existence theorems is automatically satisfied in the C∞ case, so no additional assumptions are needed beyond ∞-nonresonance.