Stochastic Asymptotic Phase

• Stochastic Asymptotic Phase is a spectral extension of deterministic phase, defined as the argument of the dominant eigenfunction from the stochastic Koopman generator.
• It employs the backward Kolmogorov operator to relate noisy oscillator dynamics to a uniform, ensemble-averaged phase progression.
• It enables practical phase–amplitude reduction using numerical eigenanalysis and data-driven methods like Extended Dynamic Mode Decomposition.

Stochastic asymptotic phase is a spectral generalization of deterministic asymptotic phase for noisy oscillators. In the formulation based on the backward Kolmogorov operator, or equivalently the stochastic Koopman generator, it is defined as the argument of the eigenfunction associated with the dominant complex eigenvalue of the stochastic dynamics. If $Q_1(x)$ is the eigenfunction corresponding to the leading complex eigenvalue $\mu_1+i\omega_1$, then the stochastic asymptotic phase is $\Theta(x)=\arg Q_1(x)$; under the standard assumptions of a stationary distribution, a unique dominant complex conjugate pair, and a spectral gap, the ensemble-averaged phase advances linearly with frequency $\omega_1$. This construction extends phase from noise-perturbed limit cycles to strongly stochastic and even noise-induced oscillations, while preserving a direct link to operator spectra, isochrons, and phase reduction (Thomas et al., 2014, Kato et al., 2021, Takata et al., 16 Jan 2025).

1. Deterministic antecedents and Koopman reinterpretation

For a smooth deterministic system

$$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)),\qquad \mathbf{x}(t)\in\mathbb{R}^N,$$

with an exponentially stable periodic orbit $\mathbf{x}_0(t)$ of period $T$, frequency $\omega=2\pi/T$, and Floquet exponents $\lambda_m$ with $\mathrm{Re}\,\lambda_m<0$, the asymptotic phase function $\mu_1+i\omega_1$0 is defined on the basin of attraction $\mu_1+i\omega_1$1 by

$\mu_1+i\omega_1$2

Along any trajectory in $\mu_1+i\omega_1$3, the phase therefore satisfies

$\mu_1+i\omega_1$4

so the phase increases at constant angular velocity. The level sets of $\mu_1+i\omega_1$5 are the isochrons: states that asymptotically converge to the same point on the limit cycle and share the same asymptotic timing. In parallel, amplitude functions $\mu_1+i\omega_1$6 are defined by

$\mu_1+i\omega_1$7

and their level sets are isostables. The collection $\mu_1+i\omega_1$8 yields a nonlinear coordinate system in which the deterministic dynamics are globally linearized inside the basin: $\mu_1+i\omega_1$9 These coordinates underlie deterministic phase reduction and phase–amplitude reduction (Takata et al., 16 Jan 2025).

Koopman operator theory recasts the same structure spectrally. If $\Theta(x)=\arg Q_1(x)$0 is the Koopman operator and $\Theta(x)=\arg Q_1(x)$1 its generator, then

$\Theta(x)=\arg Q_1(x)$2

satisfies

$\Theta(x)=\arg Q_1(x)$3

while each $\Theta(x)=\arg Q_1(x)$4 satisfies

$\Theta(x)=\arg Q_1(x)$5

Hence the deterministic asymptotic phase is the argument of a Koopman eigenfunction, and deterministic amplitudes are Koopman eigenfunctions associated with Floquet exponents. This observation is the basis for the stochastic generalization (Kato et al., 2021).

2. Spectral definition for stochastic oscillators

For an Itô SDE

$\Theta(x)=\arg Q_1(x)$6

the forward and backward Fokker–Planck operators are adjoint, and the stochastic Koopman operator

$\Theta(x)=\arg Q_1(x)$7

has infinitesimal generator

$\Theta(x)=\arg Q_1(x)$8

with $\Theta(x)=\arg Q_1(x)$9. Thus stochastic Koopman eigenfunctions are eigenfunctions of the backward Fokker–Planck operator (Takata et al., 16 Jan 2025).

The spectral definition of stochastic asymptotic phase assumes that the dominant nontrivial eigenvalues form a complex conjugate pair

$\omega_1$0

with $\omega_1$1. If $\omega_1$2 is the eigenfunction satisfying

$\omega_1$3

then the stochastic asymptotic phase is defined by

$\omega_1$4

The associated averaged phase is not defined pointwise along a single noisy trajectory; instead, for $\omega_1$5,

$\omega_1$6

satisfies

$\omega_1$7

and therefore

$\omega_1$8

In this sense the stochastic asymptotic phase is a coordinate whose dominant oscillatory Koopman mode rotates at constant angular frequency in expectation (Kato et al., 2021, Takata et al., 16 Jan 2025).

The original operator-based definition characterized a system as robustly oscillatory when the first nontrivial eigenvalue is complex, oscillatory coherence is strong, and the leading complex pair is spectrally isolated from the rest of the spectrum. In that regime, the slowest decaying oscillatory mode dominates the long-time relaxation of the transition density, and the argument of the corresponding backward eigenfunction serves as phase (Thomas et al., 2014).

An analytically tractable two-dimensional Ornstein–Uhlenbeck study sharpened the distinction between backward and forward phase descriptions. In that setting, the backward phase is the argument of the backward eigenfunction corresponding to the least negative eigenvalue, the forward phase is the argument of the forward eigenfunction, the isochrons of both are spokes of a wheel, the spacing of those isochrons differs between backward and forward phases, the backward phase is completely determined by the deterministic part of the vector field, the forward phase also depends on the noise matrix, and the mean progression of the backward phase in time is always uniform, whereas this is not true for the forward phase except in the rotationally symmetric case (Thomas et al., 2019).

3. Stochastic amplitudes, isostables, and effective oscillatory backbones

The amplitude counterpart of stochastic asymptotic phase is defined from the leading non-oscillatory decay mode. In the Koopman formulation, one selects a nonzero eigenvalue $\omega_1$9 of $$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)),\qquad \mathbf{x}(t)\in\mathbb{R}^N,$$0 with the largest real part that is not in the oscillatory branch, and its eigenfunction $$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)),\qquad \mathbf{x}(t)\in\mathbb{R}^N,$$1 satisfies

$$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)),\qquad \mathbf{x}(t)\in\mathbb{R}^N,$$2

The average amplitude coordinate

$$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)),\qquad \mathbf{x}(t)\in\mathbb{R}^N,$$3

then obeys

$$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)),\qquad \mathbf{x}(t)\in\mathbb{R}^N,$$4

Its level sets are stochastic isostables, and the zero-level set $$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)),\qquad \mathbf{x}(t)\in\mathbb{R}^N,$$5 can be associated with a stochastic periodic orbit or effective attractor around which the noisy trajectory fluctuates (Takata et al., 16 Jan 2025).

A closely related construction defines the stochastic isostable coordinate as the eigenfunction with the least negative nontrivial real eigenvalue of the backward Kolmogorov operator. In two dimensions, if $$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)),\qquad \mathbf{x}(t)\in\mathbb{R}^N,$$6 is that eigenfunction, then

$$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)),\qquad \mathbf{x}(t)\in\mathbb{R}^N,$$7

The set

$$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)),\qquad \mathbf{x}(t)\in\mathbb{R}^N,$$8

acts as the effective cycle toward which the mean amplitude relaxes. This completed the phase–amplitude description proposed for stochastic oscillators by pairing the dominant complex mode with the leading real decay mode (Pérez-Cervera et al., 2021).

The same framework was used to show that noise-induced oscillations can possess a well-defined phase–amplitude geometry even without an underlying deterministic limit cycle. In the excitable FitzHugh–Nagumo regime, the backward Fokker–Planck spectrum still exhibits a leading complex pair and a dominant real eigenvalue; the resulting phase field increases along a ring of minimal amplitude, and that ring is entirely noise-sustained. A plausible implication is that stochastic asymptotic phase is not merely a perturbative limit-cycle concept, but a spectral coordinate for any Markov oscillator with a dominant oscillatory mode (Kato et al., 2021).

For planar systems, an effective deterministic vector field can be defined so that the phase and isostable eigenfunctions evolve exactly with their respective eigenvalues: $$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)),\qquad \mathbf{x}(t)\in\mathbb{R}^N,$$9 In examples such as a noisy spiral sink and a noisy heteroclinic oscillator, the resulting effective flow possesses a limit cycle $\mathbf{x}_0(t)$0 even when the original deterministic vector field does not. This suggests a useful geometric interpretation of the stochastic oscillatory backbone (Pérez-Cervera et al., 2021).

4. Computation from operators and time series

The operator-theoretic definition can be implemented either by direct numerical eigenanalysis of the backward Fokker–Planck operator or by data-driven approximation of the stochastic Koopman semigroup. In direct approaches, $\mathbf{x}_0(t)$1 is discretized on a computational domain, its eigenvalues near the imaginary axis are computed, and the phase and amplitude functions are reconstructed as

$\mathbf{x}_0(t)$2

or via the modulus $\mathbf{x}_0(t)$3 for visualization. This strategy was used for the noisy Stuart–Landau oscillator, for noisy and noise-induced FitzHugh–Nagumo dynamics, and for a semiclassical quantum Stuart–Landau or van der Pol model with Kerr nonlinearity (Kato et al., 2021).

A data-driven version uses Extended Dynamic Mode Decomposition. Given a time series

$\mathbf{x}_0(t)$4

and a dictionary $\mathbf{x}_0(t)$5, one forms

$\mathbf{x}_0(t)$6

and solves

$\mathbf{x}_0(t)$7

The eigenvalues and left eigenvectors of $\mathbf{x}_0(t)$8 approximate the eigenvalues and eigenfunctions of $\mathbf{x}_0(t)$9, with

$T$0

The stochastic phase is reconstructed from the leading complex eigenpair as

$T$1

and the leading amplitude from the next-slowest non-oscillatory mode (Takata et al., 16 Jan 2025).

The noisy FitzHugh–Nagumo neuron model provided a concrete demonstration. The SDE

$T$2

was studied with

$T$3

The SDE was simulated by Euler–Maruyama with timestep $T$4, observations were sampled at interval $T$5, a single long trajectory was used, $T$6 data points were reported to suffice for accurate reconstruction and even $T$7 could be adequate, the dictionary consisted of $T$8 radial basis functions with centers obtained by k-means clustering, and the Tikhonov regularization parameter was $T$9. The