Koopman Phase–Amplitude Coordinates

• Koopman Phase–Amplitude Coordinates are intrinsic coordinates defined using Koopman eigenfunctions that decouple phase and amplitude dynamics.
• For periodic orbits, the phase coordinate arises from the eigenfunction's argument (iω) while amplitude coordinates derive from stable Floquet exponents, enabling uniform phase advancement and exponential decay.
• These coordinates facilitate a global linearization of unforced dynamics on attraction basins, ensuring unique semiconjugacies under proper regularity and nonresonance conditions.

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 (Kvalheim et al., 2019, Mauroy et al., 2018).

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 $\phi^t$0, differentiation at $\phi^t$1 yields the transport PDE

$\phi^t$2

This representation shifts the dynamics from nonlinear state evolution to linear evolution of observables (Kvalheim et al., 2019).

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