Koopman Eigenfunctions in Dynamical Systems

• Koopman eigenfunctions are special observables that linearize nonlinear dynamics by providing intrinsic coordinates for spectral expansions and data-driven methodologies.
• They are constructed via limit processes under spectral spread conditions, ensuring the existence and uniqueness of principal eigenfunctions near attracting fixed points or periodic orbits.
• The algebra generated by these eigenfunctions enables a global linearization framework, where observables evolve diagonally and isostable coordinates are rigorously defined.

Koopman eigenfunctions are special observables for dynamical systems that provide a rigorous linearization tool via the spectral theory of the associated Koopman operator. They serve as global, intrinsic coordinates in which both nonlinear finite- and infinite-dimensional dynamics can be described by (often diagonal) linear evolution equations, admitting spectral expansions and enabling powerful analytical and data-driven methodologies.

1. Definitions and Fundamental Properties

Given a smooth dynamical system $\Phi^t: Q \to Q$ (either discrete or continuous time) on a manifold $Q$, the Koopman operator acts on observables $\phi: Q \to \mathbb{C}$ by composition (pullback): $(U^t \phi)(x) = \phi(\Phi^t(x))$ A nonzero observable $\phi$ is a Koopman eigenfunction with exponent $\mu\in\mathbb{C}$ if

$$\phi(\Phi^t(x)) = e^{\mu t} \phi(x) \qquad \forall t$$

For discrete-time $t\in\mathbb{Z}$, one writes $K\phi = \lambda\phi$ with $\lambda = e^{\mu}$.

Koopman eigenfunctions are a special case of linearizing semiconjugacies: for $m=1$, one has $\psi\circ\Phi^t = e^{\mu t} \psi$. More generally, for $m>1$, a smooth $\psi: Q\to\mathbb{C}^m$ and $A\in\mathbb{C}^{m\times m}$ can provide a semiconjugacy $\psi\circ\Phi^t = e^{tA} \psi$ linking the nonlinear flow to linear evolution in $\mathbb{C}^m$.

2. Existence, Uniqueness, and Construction of Principal Eigenfunctions

For a $C^1$ system $\Phi$ defined on the basin $Q$ of an attracting hyperbolic fixed point $x_0$ or a periodic orbit, the existence and uniqueness of "principal" Koopman eigenfunctions—those associated with the eigenvalues of the linearized system at $x_0$ or the periodic orbit—are established under a spectral spread condition

$$\nu(e^A, D_{x_0}\Phi^1) = \max_{\mu, \lambda} \ln|\mu|/\ln|\lambda| < k+\alpha$$

for exponents $k\in\mathbb{N}\cup\{\infty\}$ and regularity $\alpha\in[0,1]$, and a $k$-nonresonance requirement preventing polynomial-type resonances between nonlinear terms and the spectral data.

Principal eigenfunctions can be constructed as limits: $\psi(x) = \lim_{t\to\infty} e^{-tA}P(\Phi^t(x))$ where $P$ is an approximately linearizing polynomial, and the limit exists due to contraction properties in the basin. The same strategy applies to attracting hyperbolic periodic orbits using the Floquet normal form.

Uniqueness holds: any two $C^{k,\alpha}_{loc}$ eigenfunctions normalized by the same derivative at the fixed point coincide.

3. Classification and Algebraic Structure

If $\Phi$ is $C^\infty$ and the linearization is semisimple and (infinitely) nonresonant, all $C^\infty$ Koopman eigenfunctions on the basin are finite linear combinations of monomials in the $n$ principal eigenfunctions and their conjugates: $$\phi(z) = \sum_{|\ell|+|m|\leq K} c_{\ell,m}\, \psi^{[\ell]}\bar{\psi}^{[m]}$$ with $$\psi^{[\ell]} = \psi_1^{\ell_1}\cdots\psi_n^{\ell_n}$$. For periodic orbits, integer powers of the angular eigenfunction (the "phase") are also included.

The algebra generated by the principal eigenfunctions (the "principal algebra") is sufficient to span all $C^\infty$ eigenfunctions in the basin under the structural hypotheses.

4. Isostables, Isostable Coordinates, and the Principal Algebra

Isostables are the level sets of the principal Koopman eigenfunction corresponding to the eigenvalue with the slowest nonzero real part (i.e., the "least stable" mode). Explicitly, for eigenfunction $\psi_1$, isostables are given by $\{| \psi_1(x) | = c\}$ for $c > 0$. These sets form smooth foliations of the basin and provide coordinates that quantify the asymptotic approach rate to the attractor.

Isostable coordinates generalize this construction: in higher dimensions, the non-torsion part of the system is coordinatized by the monomials in principal eigenfunctions, and for periodic orbits, the eigenfunction $\psi_\theta$ on $S^1$ gives the phase. The slow isostable can be rigorously constructed and is unique in $C^{1,\alpha}_{loc}$ for $C^2$ systems (Kvalheim et al., 2019).

The principal algebra $A^{k,\alpha}_\Phi$ is the subalgebra generated by all principal eigenfunctions. This algebra is uniquely determined by the dynamics (not by any choice of conjugacy) when the global Sternberg or Floquet conjugacy exists in $C^{k,\alpha}_{loc}$.

5. Pullback Algebras and Uniqueness

Mohr & Mezić (Mohr et al., 2016) defined the principal algebra $A_{D\Phi^1}$ for the linearized system and then considered its pullback to the nonlinear system via a topological conjugacy. Under the conditions of global $C^{k,\alpha}_{loc}$ Sternberg (for fixed points) or Floquet (for periodic orbits) conjugacy, this pullback algebra becomes canonical and uniquely coincides with the principal algebra $A^{k,\alpha}_\Phi$ of the nonlinear system (Kvalheim et al., 2019). Thus, the spectral expansion in terms of Koopman eigenfunctions and their algebra holds canonically in a neighborhood of the attractor, not depending on any artificial choice.

The limiting process used in computational works (e.g., for finding "faster" isostable coordinates as in [wilson2018greater, monga2019phase]) is made rigorous: the convergence of the coordinate-defining limit is guaranteed precisely when the (quantifiable) spectral spread condition is satisfied.

6. Implications for Spectral Expansions and Global Linearization

The principal algebra generated by the finite set of principal eigenfunctions enables the spectral expansion of any smooth observable (vanishing at the attractor). In precise terms:

• Any $C^\infty$ observable can be written as a convergent sum of monomials in principal eigenfunctions and their conjugates.
• Time evolution of all such observables is diagonalized: monomials in principal eigenfunctions evolve with exponential factors $e^{(\ell\lambda + m\bar{\lambda})t}$.
• The analytic conjugacy straightens the nonlinear dynamics to the linear flow, and principal eigenfunctions correspond precisely to the normal coordinates under this conjugacy.

Thus, the Koopman eigenspectrum provides a basis of global coordinates (on the basin of attraction) in which the nonlinear system is fully linearized. The structure of isostables and isochrons for periodic orbits is directly linked to the geometry of these eigenfunctions, and the global uniqueness of the principal algebra eliminates ambiguities in the functional model for dissipative systems.

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References:

• "Koopman principle eigenfunctions and linearization of diffeomorphisms" (Mohr et al., 2016)
• "Existence and uniqueness of global Koopman eigenfunctions for stable fixed points and periodic orbits" (Kvalheim et al., 2019)