Isostable Framework for Nonlinear Dynamics

• The isostable coordinate framework is a nonlinear dynamics description that uses slowest decaying Koopman eigenfunctions or Floquet modes to organize state-space evolution.
• It defines invariant slow manifolds and underpins reduced-order models, enabling data-driven inference and effective handling of forcing in fixed-point and oscillatory systems.
• Advanced numerical strategies, including predictor–corrector methods and asymptotic expansions, address backward integration instability and stiffness in these high-order reductions.

Searching arXiv for the specified papers to ground the article in current metadata and citations. arxiv_search.query({"^^^^^^^^12search_query12 OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^^^^^"," OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12^^^^,"^^^^^^^^12sort_by12 arxiv_search.query({"^^^^^^^^12search_query12 and Computation of Slow Manifolds Using the Isostable Coordinate System\" OR ^^^^^^^^12ti:\12 Inference of High-Accuracy Isostable-Based Dynamical Models in Response to External Inputs\" OR ^^^^^^^^12ti:\12 for stochastic oscillators\" OR ^^^^^^^^12ti:\12 into oscillator network dynamics using a phase-isostable framework\"","^^^^^^^^12max_results12 OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12^^^^,"^^^^^^^^12sort_by12 The isostable coordinate framework is a coordinate description of nonlinear dynamics in which state-space evolution is organized by asymptotic decay rates associated with Koopman eigenfunctions or, for periodic orbits, Floquet modes. For systems with a stable fixed point, the framework focuses on the slowest decaying principal Koopman eigenmodes and defines scalar coordinates whose level sets are isostables; for stochastic oscillators and deterministic limit cycles, closely related constructions produce amplitude-like coordinates complementary to asymptotic phase. In the formulations developed for fixed-point attractors, external-input reductions, stochastic oscillators, and oscillator networks, isostable coordinates provide a common language for invariant geometry, reduced-order modeling, and bifurcation analysis (&&&^^^^12search_query12^^^^&&&, &&&^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^&&&, &&&^^^^12max_results12^^^^&&&, &&&^^^^12sort_by12^^^^&&&).

^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^. Koopman-theoretic definition and local structure

For a smooth autonomous ODE

PRESERVED_PLACEHOLDER_^^^^12search_query12^^^^

with hyperbolic fixed point PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^, the Koopman operator acts on observables PRESERVED_PLACEHOLDER_^^^^12max_results12^^^^ by

PRESERVED_PLACEHOLDER_^^^^12sort_by12^^^^

where PRESERVED_PLACEHOLDER_^^^^12relevance12^^^^ is the flow. Although the state-space dynamics are nonlinear, PRESERVED_PLACEHOLDER_^^^^12sort_order12^^^^ is linear but generally infinite-dimensional. A Koopman eigenfunction PRESERVED_PLACEHOLDER_^^^^12descending12^^^^ and eigenvalue PRESERVED_PLACEHOLDER_^^^^12search_query12^^^^ satisfy

PRESERVED_PLACEHOLDER_^^^^12ti:\12^^^^

When the Jacobian PRESERVED_PLACEHOLDER_^^^^12 OR ti:\12^^^^ has eigenvalues ordered by

PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12^^^^

the associated PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^ are the principal eigenfunctions, and the real part or modulus of PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12max_results12^^^^ yields an isostable coordinate measuring distance along the PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_by12^^^^-th mode (&&&^^^^12search_query12^^^^&&&).

A constructive limit definition is obtained by fixing left and right eigenvectors PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12relevance12^^^^ of PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_order12^^^^, normalized so that PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12descending12^^^^, and setting

PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12^^^^

Its level sets PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12ti:\12^^^^ are the isostables associated with decay rate PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12 OR ti:\12^^^^. Differentiation along trajectories gives

PRESERVED_PLACEHOLDER_^^^^12max_results12search_query12^^^^

so each principal isostable decays or grows exponentially at rate PRESERVED_PLACEHOLDER_^^^^12max_results12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^ (&&&^^^^12search_query12^^^^&&&).

A closely related normalization appears in isostable reduction for forced systems. There, Koopman eigenfunctions PRESERVED_PLACEHOLDER_^^^^12max_results12max_results12^^^^ satisfy the spectral PDE

PRESERVED_PLACEHOLDER_^^^^12max_results12sort_by12^^^^

with normalization

PRESERVED_PLACEHOLDER_^^^^12max_results12relevance12^^^^

and the real-valued isostable coordinate PRESERVED_PLACEHOLDER_^^^^12max_results12sort_order12^^^^ is defined by filtering out all but the slowest decaying Koopman mode through a limit of the same form (&&&^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^&&&). Near the fixed point, these definitions recover the linear eigendirections because PRESERVED_PLACEHOLDER_^^^^12max_results12descending12^^^^ locally (&&&^^^^12search_query12^^^^&&&).

^^^^12max_results12^^^^. Slow manifolds as zero sets of fast isostables

When there is a pronounced spectral gap at index PRESERVED_PLACEHOLDER_^^^^12max_results12search_query12^^^^,

PRESERVED_PLACEHOLDER_^^^^12max_results12ti:\12^^^^

the framework defines a PRESERVED_PLACEHOLDER_^^^^12max_results12 OR ti:\12^^^^-dimensional slow manifold by setting all faster isostable amplitudes to zero:

PRESERVED_PLACEHOLDER_^^^^12sort_by12search_query12^^^^

This identifies the slow manifold as the set on which the fastest decaying principal Koopman coordinates vanish (&&&^^^^12search_query12^^^^&&&).

The definition immediately implies invariance. On PRESERVED_PLACEHOLDER_^^^^12sort_by12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^, the coordinates PRESERVED_PLACEHOLDER_^^^^12sort_by12max_results12^^^^ are identically zero, so their derivatives vanish under PRESERVED_PLACEHOLDER_^^^^12sort_by12sort_by12^^^^, and the flow remains on PRESERVED_PLACEHOLDER_^^^^12sort_by12relevance12^^^^. Near PRESERVED_PLACEHOLDER_^^^^12sort_by12sort_order12^^^^, the manifold reduces to PRESERVED_PLACEHOLDER_^^^^12sort_by12descending12^^^^, the span of the slow eigenvectors. Away from the linear regime, PRESERVED_PLACEHOLDER_^^^^12sort_by12search_query12^^^^ is a nonlinear manifold of codimension PRESERVED_PLACEHOLDER_^^^^12sort_by12ti:\12^^^^, locally transverse to the fast eigendirections (&&&^^^^12search_query12^^^^&&&).

This formulation is not merely a local tangent-space statement. The motivating point of the construction is that the condition PRESERVED_PLACEHOLDER_^^^^12sort_by12 OR ti:\12^^^^ remains meaningful beyond the linear neighborhood of the attractor, so the same coordinate language used for spectral decomposition also gives a geometric definition of the slow manifold in the nonlinear regime (&&&^^^^12search_query12^^^^&&&). This suggests a direct bridge between Koopman spectral objects and model order reduction: the reduced dynamics are obtained by retaining the slowest coordinates and eliminating fast amplitudes by an invariant constraint.

^^^^12sort_by12^^^^. Backward-time computation and numerical strategies

Direct backward integration of the original ODE from a point on PRESERVED_PLACEHOLDER_^^^^12relevance12search_query12^^^^ is numerically unstable once the trajectory leaves a small neighborhood of PRESERVED_PLACEHOLDER_^^^^12relevance12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^, because any small error in a fast isostable PRESERVED_PLACEHOLDER_^^^^12relevance12max_results12^^^^ blows up like PRESERVED_PLACEHOLDER_^^^^12relevance12sort_by12^^^^. To avoid this, the framework rewrites the backward dynamics in isostable coordinates. With backward-time variable PRESERVED_PLACEHOLDER_^^^^12relevance12relevance12^^^^, one has

PRESERVED_PLACEHOLDER_^^^^12relevance12sort_order12^^^^

where PRESERVED_PLACEHOLDER_^^^^12relevance12descending12^^^^. Stacking PRESERVED_PLACEHOLDER_^^^^12relevance12search_query12^^^^ into a matrix and enforcing PRESERVED_PLACEHOLDER_^^^^12relevance12ti:\12^^^^ on PRESERVED_PLACEHOLDER_^^^^12relevance12 OR ti:\12^^^^ yields

PRESERVED_PLACEHOLDER_^^^^12sort_order12search_query12^^^^

If PRESERVED_PLACEHOLDER_^^^^12sort_order12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^ and PRESERVED_PLACEHOLDER_^^^^12sort_order12max_results12^^^^ are known along PRESERVED_PLACEHOLDER_^^^^12sort_order12sort_by12^^^^, this evolution marches backward without exciting fast modes (&&&^^^^12search_query12^^^^&&&).

For the slow coordinates PRESERVED_PLACEHOLDER_^^^^12sort_order12relevance12^^^^, the isostable gradients satisfy the adjoint variational equation

PRESERVED_PLACEHOLDER_^^^^12sort_order12sort_order12^^^^

with initial condition PRESERVED_PLACEHOLDER_^^^^12sort_order12descending12^^^^. Because these are slow modes, backward integration remains accurate over substantial intervals. The fast gradients PRESERVED_PLACEHOLDER_^^^^12sort_order12search_query12^^^^, however, are not computed directly. Instead, the computation uses their duals PRESERVED_PLACEHOLDER_^^^^12sort_order12ti:\12^^^^, which solve

PRESERVED_PLACEHOLDER_^^^^12sort_order12 OR ti:\12^^^^

with bi-orthonormality PRESERVED_PLACEHOLDER_^^^^12descending12search_query12^^^^. Since the backward formula only needs the subspace spanned by PRESERVED_PLACEHOLDER_^^^^12descending12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^, it is enough to approximate the orthogonal complement of PRESERVED_PLACEHOLDER_^^^^12descending12max_results12^^^^ (&&&^^^^12search_query12^^^^&&&).

Two strategies are developed. The asymptotic-expansion approach expands PRESERVED_PLACEHOLDER_^^^^12descending12sort_by12^^^^ and PRESERVED_PLACEHOLDER_^^^^12descending12relevance12^^^^ in Taylor series in the isostables near PRESERVED_PLACEHOLDER_^^^^12descending12sort_order12^^^^, for example

PRESERVED_PLACEHOLDER_^^^^12descending12descending12^^^^

and then matches with PRESERVED_PLACEHOLDER_^^^^12descending12search_query12^^^^ to solve for PRESERVED_PLACEHOLDER_^^^^12descending12ti:\12^^^^ to arbitrary order. The method works well in low dimension or mild nonlinearity but becomes unwieldy beyond PRESERVED_PLACEHOLDER_^^^^12descending12 OR ti:\12^^^^–PRESERVED_PLACEHOLDER_^^^^12search_query12search_query12^^^^-th order. The predictor–corrector approach instead uses the approximation PRESERVED_PLACEHOLDER_^^^^12search_query12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^, integrates backward using the slow gradients, then corrects the resulting point by solving for a displacement PRESERVED_PLACEHOLDER_^^^^12search_query12max_results12^^^^ in the fast eigendirections so that the exact isostable-gradient condition is restored (&&&^^^^12search_query12^^^^&&&).

The reported numerical issues are threefold. Timescale separation makes direct backward integration exponentially unstable in fast modes; the reformulated backward equation avoids this by keeping PRESERVED_PLACEHOLDER_^^^^12search_query12sort_by12^^^^. Stiffness arises in the adjoint equation when PRESERVED_PLACEHOLDER_^^^^12search_query12relevance12^^^^ is very negative, motivating a stiff integrator or small steps. Non-uniqueness requires choosing PRESERVED_PLACEHOLDER_^^^^12search_query12sort_order12^^^^ so that there are no resonances PRESERVED_PLACEHOLDER_^^^^12search_query12descending12^^^^ for slow indices; this guarantees convergence of the asymptotic expansions (&&&^^^^12search_query12^^^^&&&).

^^^^12relevance12^^^^. Reduced-order models with forcing and data-driven inference

For systems with a stable fixed point driven by an input PRESERVED_PLACEHOLDER_^^^^12search_query12search_query12^^^^ entering through PRESERVED_PLACEHOLDER_^^^^12search_query12ti:\12^^^^, an PRESERVED_PLACEHOLDER_^^^^12search_query12 OR ti:\12^^^^-mode isostable reduction keeps only the slowest coordinates PRESERVED_PLACEHOLDER_^^^^12ti:\12search_query12^^^^ and represents the reduced dynamics and output as

PRESERVED_PLACEHOLDER_^^^^12ti:\12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^

Here PRESERVED_PLACEHOLDER_^^^^12ti:\12max_results12^^^^ and PRESERVED_PLACEHOLDER_^^^^12ti:\12sort_by12^^^^ are expanded in multivariate Taylor series in PRESERVED_PLACEHOLDER_^^^^12ti:\12relevance12^^^^, and truncating at total degree PRESERVED_PLACEHOLDER_^^^^12ti:\12sort_order12^^^^ yields a PRESERVED_PLACEHOLDER_^^^^12ti:\12descending12^^^^-th-order accurate reduced model (&&&^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^&&&).

When the underlying equations are unknown, the coefficients can be inferred from steady-state responses to sinusoidal probing. With rank-one input

PRESERVED_PLACEHOLDER_^^^^12ti:\12search_query12^^^^

the steady-state output admits an PRESERVED_PLACEHOLDER_^^^^12ti:\12ti:\12^^^^-expansion and Fourier series

PRESERVED_PLACEHOLDER_^^^^12ti:\12 OR ti:\12^^^^

The key structural fact is that the PRESERVED_PLACEHOLDER_^^^^12 OR ti:\12search_query12^^^^-th harmonic first appears at PRESERVED_PLACEHOLDER_^^^^12 OR ti:\12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^, and its amplitude is a linear combination of the PRESERVED_PLACEHOLDER_^^^^12 OR ti:\12max_results12^^^^-th-order Taylor coefficients in PRESERVED_PLACEHOLDER_^^^^12 OR ti:\12sort_by12^^^^ and PRESERVED_PLACEHOLDER_^^^^12 OR ti:\12relevance12^^^^. At first order,

PRESERVED_PLACEHOLDER_^^^^12 OR ti:\12sort_order12^^^^

with PRESERVED_PLACEHOLDER_^^^^12 OR ti:\12descending12^^^^ collecting measured first harmonics, PRESERVED_PLACEHOLDER_^^^^12 OR ti:\12search_query12^^^^ known analytically in terms of PRESERVED_PLACEHOLDER_^^^^12 OR ti:\12ti:\12^^^^, and PRESERVED_PLACEHOLDER_^^^^12 OR ti:\12 OR ti:\12^^^^ containing the unknown coefficient products. Pseudoinverse solution gives

PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12search_query12^^^^

Second and higher orders have the same structure, with remainder terms depending only on lower-order quantities. The eigenvalues PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^ can also be refined by Newton iteration on the first-order relation (&&&^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^&&&).

The algorithmic summary consists of a preliminary data-driven estimate of PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12max_results12^^^^ from delay-embedded outputs and POD, symbolic assembly of response matrices PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12sort_by12^^^^, harmonic extraction at probe frequencies PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12relevance12^^^^, pseudoinverse recovery of the coefficient vectors PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12sort_order12^^^^, and optional Newton refinement before assembling the final reduced model (&&&^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^&&&).

The examples emphasize accuracy under finite forcing amplitudes. In a two-dimensional test model with PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12descending12^^^^ and PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12search_query12^^^^, one isostable suffices, the third-order reduction matches the full dynamics perfectly when PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12ti:\12^^^^ and PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12 OR ti:\12^^^^ are known, and data-driven inference from noisy output recovers PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12^^^^ together with coefficients up to cubic order. For a population of PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^ synaptically coupled Morris–Lecar-type neurons, two complex-conjugate isostables suffice, and the second-order inferred model outperforms the linear one by an order of magnitude in mean absolute error under composite transient inputs. For the one-dimensional Burgers’ equation projected onto five POD modes, a second-order three-isostable model reduces PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12max_results12^^^^-error by two orders of magnitude compared to the first-order model under composite multi-frequency boundary inputs (&&&^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^&&&).

A related input-driven reduction appears in the Goodwin oscillator example used for slow-manifold computation. After adding an external input PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_by12^^^^ to the PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12relevance12^^^^-equation and restricting to the manifold PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_order12^^^^ with PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12descending12^^^^, the reduced model becomes

PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12^^^^

where PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12ti:\12^^^^ is tabulated along the computed manifold. In that example, the reduced one-complex-dimensional ODE reproduces steady-state periodic and period-doubling behavior far beyond the linear approximation; the full and isostable-reduced models both show a period-doubling bifurcation at input amplitude PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12 OR ti:\12^^^^–PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12max_results12search_query12^^^^, whereas a naive linearization fails to capture the bifurcation, and the maximum amplitude of PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12max_results12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^ versus PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12max_results12max_results12^^^^ matches to within a few percent (&&&^^^^12search_query12^^^^&&&).

^^^^12sort_order12^^^^. Stochastic and phase–amplitude extensions

For stochastic oscillators governed by the Itô SDE

PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12max_results12sort_by12^^^^

with diffusion tensor PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12max_results12relevance12^^^^, the relevant operator is the backward Kolmogorov, or stochastic Koopman, operator

PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12max_results12sort_order12^^^^

Under standard ellipticity and suitable boundary conditions, PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12max_results12descending12^^^^ and PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12max_results12search_query12^^^^ admit discrete spectra and biorthogonal eigenbases. A system is termed robustly oscillatory when the nonzero eigenvalue of PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12max_results12ti:\12^^^^ with maximal real part is a complex conjugate pair PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12max_results12 OR ti:\12^^^^, with PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_by12search_query12^^^^, PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_by12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^, and all other eigenvalues satisfy PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_by12max_results12^^^^. The asymptotic phase eigenfunction is then PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_by12sort_by12^^^^, and the asymptotic phase is

PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_by12relevance12^^^^

If there is also a unique real eigenvalue PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_by12sort_order12^^^^ with maximal real part among real eigenvalues, the corresponding real eigenfunction PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_by12descending12^^^^ is interpreted as the stochastic isostable coordinate (&&&^^^^12max_results12^^^^&&&).

Normalization is fixed by biorthogonality, and PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_by12search_query12^^^^ is typically shifted so that its zero-level set defines an effective limit cycle

PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_by12ti:\12^^^^

In the phase–amplitude variables

PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_by12 OR ti:\12^^^^

Itô’s formula yields

PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12relevance12search_query12^^^^

PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12relevance12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^

Thus, in the mean sense,

PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12relevance12max_results12^^^^

The noisy linear focus provides a closed-form example: for the planar Ornstein–Uhlenbeck process with

PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12relevance12sort_by12^^^^

one obtains

PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12relevance12relevance12^^^^

PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12relevance12sort_order12^^^^

so the effective limit cycle is the circle PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12relevance12descending12^^^^ (&&&^^^^12max_results12^^^^&&&).

For deterministic limit cycles, the analogous construction uses Floquet theory rather than Koopman eigenfunctions at a fixed point. If the uncoupled ODE admits a stable PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12relevance12search_query12^^^^-periodic orbit PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12relevance12ti:\12^^^^ with dominant nontrivial Floquet exponent PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12relevance12 OR ti:\12^^^^, the asymptotic phase PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_order12search_query12^^^^ is defined by PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_order12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^, and the slowest-decaying transverse coordinate is an isostable PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_order12max_results12^^^^ satisfying

PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_order12sort_by12^^^^

Near the cycle, the phase–isostable normal form under weak perturbation PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_order12relevance12^^^^ is

PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_order12sort_order12^^^^

PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_order12descending12^^^^

where PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_order12search_query12^^^^ is the iPRC and PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_order12ti:\12^^^^ is the iIRC (&&&^^^^12sort_by12^^^^&&&). The phase and isostable coordinates therefore separate tangential timing from the slowest transverse amplitude decay.

^^^^12descending12^^^^. Network formulations, comparative scope, and limitations

For PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12sort_order12 OR ti:\12^^^^ identical coupled oscillators

PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12descending12search_query12^^^^

retaining the phase PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12descending12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^ and a single slow isostable PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12descending12max_results12^^^^ at each node gives, after Taylor expansion and first-order averaging in PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12descending12sort_by12^^^^,

PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12descending12relevance12^^^^

PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12descending12sort_order12^^^^

The six PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12descending12descending12^^^^-periodic coupling functions PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12descending12search_query12^^^^ are defined by averaging PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12descending12ti:\12^^^^ over one period and depend on PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12descending12 OR ti:\12^^^^, the coupling Jacobians, and the Floquet eigenfunction PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12search_query12^^^^ (&&&^^^^12sort_by12^^^^&&&).

The reduced network equations support explicit existence and stability conditions for phase-locked states. For a PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^ phase-locked solution PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12max_results12^^^^ with constant PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12sort_by12^^^^, the locked amplitudes satisfy PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12relevance12^^^^ and the collective frequency shift satisfies

PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12sort_order12^^^^

Linearization produces a PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12descending12^^^^ Jacobian in block form PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12search_query12^^^^, and stability requires all eigenvalues except the rotational zero mode to have negative real part. In the globally coupled case, synchrony reduces to a PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12ti:\12^^^^ matrix PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12search_query12 OR ti:\12^^^^, so stability requires

PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12ti:\12search_query12^^^^

Taking PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12ti:\12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^ recovers the classical phase-only criterion PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12ti:\12max_results12^^^^ (&&&^^^^12sort_by12^^^^&&&).

The mean-field complex Ginzburg–Landau equation provides an analytic benchmark. There, the phase-isostable reduction reproduces the synchrony boundary exactly, yields compact analytic expressions for the splay and antisynchrony boundaries, and tracks the true bifurcation loci qualitatively over a wide parameter range, whereas second- and third-order phase reductions agree only locally near PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12ti:\12sort_by12^^^^. It also predicts bistability regions between synchrony and splay in agreement with the full system. In globally coupled Morris–Lecar networks, the PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12ti:\12relevance12^^^^-dimensional phase-isostable equations predict loss and restabilization of synchrony, a narrow window of stable antisynchrony, off-invariant-manifold phase-locked states, quasiperiodic solutions born in Hopf bifurcations, and cluster states for PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12ti:\12sort_order12^^^^, while the first-order phase model predicts only trivial stable antisynchrony and unstable synchrony (&&&^^^^12sort_by12^^^^&&&).

Across the fixed-point, forced, stochastic, and network settings, several limitations are explicit. Higher-order Taylor inference suffers from combinatorial growth in the number of coefficients and from noise sensitivity because fitting divides by PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12ti:\12descending12^^^^; careful choice of PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12ti:\12search_query12^^^^ can mitigate this. The data-driven fixed-point framework requires a stable fixed point and forcing experiments that remain inside its basin of attraction. In slow-manifold computation, stiffness and fast-mode instability constrain backward marching, and asymptotic expansions require a nonresonance condition on slow indices. In network reduction, higher-order pure phase reductions become cumbersome for large PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12ti:\12ti:\12^^^^, whereas the phase-isostable model remains compact by retaining one extra coordinate per node and six coupling functions PRESERVED_PLACEHOLDER_^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12ti:\12 OR ti:\12^^^^ (&&&^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^&&&, &&&^^^^12search_query12^^^^&&&, &&&^^^^12sort_by12^^^^&&&).

Taken together, these formulations define the isostable coordinate framework as a family of reductions in which the dominant decay structure of a nonlinear system is represented explicitly. At fixed points, the coordinates identify invariant slow manifolds by the condition that fast isostables vanish; under forcing, they support high-order reduced models and purely data-driven inference; in stochastic oscillators, they supply an amplitude variable dual to asymptotic phase; and in oscillator networks, they extend phase reduction by tracking slow transverse deviations from the attracting cycle (&&&^^^^12search_query12^^^^&&&, &&&^^^^12id:(Wilson, 18 Jul 2025) OR id:(Wilson, 2021) OR id:(Pérez-Cervera et al., 2021) OR id:(Nicks et al., 2023)12^^^^&&&, &&&^^^^12max_results12^^^^&&&, &&&^^^^12sort_by12^^^^&&&).