---
title: Koopman Phase–Amplitude Coordinates
url: https://www.emergentmind.com/topics/koopman-phase-amplitude-coordinates
type: topic
---

# Koopman Phase–Amplitude Coordinates

Koopman phase–amplitude coordinates are intrinsic coordinates on the basin of attraction of a stable fixed point or a hyperbolic attracting periodic orbit obtained from Koopman eigenfunctions of the flow. In the periodic-orbit case, the phase coordinate is the argument of the eigenfunction associated with the purely imaginary eigenvalue \(i\omega\), while the amplitude coordinates are eigenfunctions associated with stable Floquet exponents; in the fixed-point case, only amplitude coordinates remain. These coordinates linearize the unforced dynamics at the level of observables, so that phase advances uniformly and amplitudes decay exponentially, and under appropriate regularity, nonresonance, and normalization conditions the resulting semiconjugacies exist globally on the entire basin and are unique [1911.11996][1803.07379].

## 1. Koopman-operator formulation

Consider the flow \(\phi^t\) generated by the ODE \(x' = f(x)\) on a smooth manifold \(Q\). The Koopman operator \(U^t\) acts on observables \(g:Q\to\mathbb{C}\) by composition,
\[
U^t g = g \circ \phi^t.
\]
A Koopman eigenfunction \(g\) with eigenvalue \(\lambda\in\mathbb{C}\) satisfies
\[
g(\phi^t(x)) = e^{\lambda t} g(x),
\]
and, for flows generated by \(x' = f(x)\), differentiation at \(t=0\) yields the transport PDE
\[
\nabla g(x)\cdot f(x)=\lambda g(x).
\]
This representation shifts the dynamics from nonlinear state evolution to linear evolution of observables [1911.11996].

A central object is a linear

Source: https://www.emergentmind.com/topics/koopman-phase-amplitude-coordinates