Discrete rotating modes and their fast/slow reduction

Definitions and mathematical statement

Suppose a planar autonomous ODE $\dot x=f(x,y),\ \dot y=g(x,y)$ has an equilibrium whose real Jacobian $A$ has eigenvalues $-\lambda\pm i\omega$, with $\lambda>0$ and $\omega\ne0$. Its noisy linearization, after translating the equilibrium to zero, is \[ dZ_t=A Z_t\,dt+\operatorname{diag}(\sigma_1,\sigma_2)\,dW_t, \] where the two coordinates of $W$ are independent standard Brownian motions. The source describes an approximation by a constant transformation, a rotation, and two independent standard Ornstein--Uhlenbeck processes. It asks for a discrete-time reaction--diffusion analogue and its relation to more general fast/slow systems. The approximation parameter, noise restrictions, error topology and reaction--diffusion model are not specified; stable complex eigenvalues alone do not state a quantitative approximation theorem.

The attached aid treats a linear finite-graph interpretation. For a finite undirected graph with symmetric Laplacian $L$, a representative coupled map is \[ U_{n+1}=(I\otimes A_0-L\otimes D)U_n+\xi_{n+1}. \] In a graph eigenbasis, a mode of eigenvalue $\nu$ has multiplier $M=A_0-\nu D$. For independent Gaussian innovations of mode covariance $Q$, a stable mode $Z_{n+1}=MZ_n+\eta_{n+1}$, $\rho(M)<1$, has stationary covariance \[ C=\sum_{j\ge0}M^jQ(M^T)^j, \qquad C=MCM^T+Q, \] and spectral density $(2\pi)^{-1}(I-Me^{-i\alpha})^{-1}Q(I-M^Te^{i\alpha})^{-1}$. Spatially independent, identically distributed vertex noise makes the graph-mode innovations independent; without a corresponding noise assumption this independence is not automatic.

For a real two-dimensional oscillatory mode with eigenvalues $re^{\pm i\theta}$, $0<r<1$, $\sin\theta\ne0$, and $Q>0$, the aid reports the exact noise-metric criterion \[ MQM^T=r^2Q. \] Precisely, this is the criterion for a stationary representation in law \[ Z_n=H R_{n\theta}V_n,\qquad V_{n+1}=rV_n+\sqrt{1-r^2}\,\zeta_{n+1}, \] where $R_\theta$ is planar rotation, $H$ is a fixed invertible matrix with $HH^T=Q/(1-r^2)$, and $V$ has two independent standard stationary scalar AR(1) coordinates; the $\zeta_n$ are independent standard Gaussian vectors. A fixed matrix transformation, rather than necessarily a scalar multiple, is allowed here.

For the fast/slow connection the aid uses an explicitly scaled linear embedding, which can be written \[ X_{n+1}=X_n+\epsilon(A_sX_n+B_fY_n),\qquad Y_{n+1}=RY_n+CX_n+\xi_{n+1},\quad \rho(R)<1, \] with centered independent innovations of covariance $Q_f$, a fixed uniform fast spectral gap, and suitable bounded second-moment initial data. Exact elimination of $Y$ produces a memory term with coefficients $B_fR^\ell C$ and colored noise. The reported averaged drift matrix and slow diffusion covariance are \[ \bar A=A_s+B_f(I-R)^{-1}C,\qquad \Sigma=B_f(I-R)^{-1}Q_f(I-R^T)^{-1}B_f^T. \] The aid reports a pointwise $L^2$ averaging error $O(\sqrt\epsilon)$, uniformly over $n\le H/\epsilon$ for fixed $H$, and a diffusion limit for the normalized fluctuation under its scaling and initial-data hypotheses. This is not a bound on the $L^2$ norm of a time supremum. In the isotropic case $R=rR_\theta$, $Q_f=qI$, $B_f=bI$, its formula becomes \[ \Sigma=\frac{qb^2}{1-2r\cos\theta+r^2}I. \] The supplied summary does not give the complete hypotheses or convergence formulation of the fluctuation theorem. These conditional linear results do not specify a general nonlinear or infinite-dimensional reaction--diffusion analogue.

Short English statement

Find a discrete reaction--diffusion counterpart of rotating noisy stable-focus dynamics and relate it to fast/slow averaging. The supplied aid gives a finite-graph linear mode analysis, an exact covariance criterion for rotating AR(1) modes, and conditional memory and diffusion formulas for a specified linear fast/slow scaling.

Sources

• AIM workshop Stochastic methods for non-equilibrium dynamical systems, Problem 13: https://aimath.org/pastworkshops/noneqdynsysproblems.pdf.
• Local registry data/unsolvedmath/problems.json, permanent ID 20001398, source code catalogue entry; canonical source aim-dynamical-systems-notes.json, notes[55].
• The record labels the statement exact and says a prior source inspection recovered the damaged extraction from page 2; this draft does not independently repeat that inspection.
• Attached machine-generated attempt 1: partial result, internal rigor label "proved", low novelty confidence. It is not peer reviewed and requires independent verification before citation; no separate literature reference is supplied.