NOTE-1020: Del Vecchio and Slotine — Contraction and singular perturbation

Contribution

Standard singular perturbation (Tikhonov, Khalil) gives O(ε) estimates for ε below an unspecified ε*, locally, and on infinite horizons only with extra stability assumptions. This note replaces the local stability hypotheses by contraction of the fast and slow subsystems and obtains global bounds with every constant explicit, valid at any ε, on the fast state’s distance from the slow manifold and on the slow state’s distance from the reduced trajectory. It extends the result to three time scales and applies it to retroactivity attenuation in a phosphorylation cycle, where the explicit constants say which parameters to tune.

Key insight

Treat the slow manifold z = γ(x(t)) as a trajectory of the fast system disturbed by the manifold’s own motion, of size ε|∂γ/∂x · f|. Contraction turns a bounded disturbance into a bounded distance, divided by the contraction rate λ_z/ε, so the fast state sits O(ε) from the manifold. The slow state is then the reduced system disturbed by f’s dependence on that O(ε) gap, and contraction of the reduced system turns it into an O(ε) trajectory error that does not accumulate.

Assumptions

• Two time scales (1): ẋ = f(x, z, t), εż = g(x, z, ε), ε ≥ 0.
• Contraction by matrix measure (Theorem 1, Table I). µ(∂f/∂x) ≤ −λ for a matrix measure µ induced by |·|₁, |·|₂, |·|_∞ or a weighted norm |Θx| with Θ a constant invertible matrix. “Contraction can use time-varying and state-dependent Θ(x, t)” is mentioned (CDC version) and never used. χ is an upper bound on the condition number of Θ.
• Partial contraction (Definition 1). In x: ẏ = f(y, z(t), t) is contracting for every z(t). In z: εẏ = g(x(t), y, ε) is contracting for every x(t) and every ε > 0, at rate λ_z/ε.
• Lemma 2: |∂γ/∂x · f(x, z, t)| ≤ d for all x, z, t, and g K-Lipschitz in ε. With f contracting in x and γ linear this cannot hold globally (f is then unbounded). The proof uses it only along the trajectory.
• Theorem 3: in addition, partial contraction in x at rate λ_x, f α-Lipschitz in z, γ α_γ-Lipschitz. As shown below, the proof needs the reduced system ẋ = f(x, γ(x), t) to be contracting, which partial contraction in x does not give.
• Section V: biomolecular systems (13)–(14) with properties P1 (TA + MB = 0, Ml = 0, MC = 0) and P2 (ker D ⊆ ker C), so that a change of variables x = Tu + My puts them in form (1) with ε = 1/G₁.
• Deterministic, continuous time throughout.

Key results

• Lemma 1 (robustness). If ẋ = f(x, t) is contracting at rate λ in a metric Θ and |d(x_p, t)| ≤ d for all x_p, t, then any trajectory of ẋ_p = f(x_p, t) + d satisfies |x_p(t) − x(t)| ≤ χe^{−λt}|x_p(0) − x(0)| + χd/λ. Proof: “With R(t) = |Θ(x_p(t) − x(t))|, one has (see [20]) dR/dt + λR ≤ |Θd(x_p, t)| … Integrating the above yields the result.” The differential inequality is cited, not shown. Re-derived below (R1).
• Theorem 2. Partial contraction in z gives a unique global root z = γ(x, ε) of g = 0; γ(x) := γ(x, 0) has ∂γ/∂x = −(∂g/∂z)⁻¹ ∂g/∂x (4), ∂g/∂z being uniformly invertible.
• Lemma 2 (fast bound). |z(t) − γ(x(t))| ≤ χ_z e^{−(λ_z/ε)t}|z(0) − γ(x(0))| + (d + K)χ_z ε/λ_z, for all t ≥ 0 (5). Re-derived (R3).
• Theorem 3 (slow bound). |x(t) − x_γ(t)| ≤ χ_x|x(0) − x_γ(0)|e^{−λ_x t} + ε[C₁(e^{−λ_x t} − e^{−(λ_z/ε)t}) + C₂(1 − e^{−λ_x t})] (7), and (8) for z − γ(x_γ), with C₁ = χ_x α χ_z |z(0) − γ(x(0))|/(λ_z − ελ_x) and C₂ = χ_x α χ_z (d + K)/(λ_z λ_x). Holds when: the reduced system is contracting at rate λ_x, not as stated (R4).
• Remark. With x(0) = x_γ(0) and ε → 0 this recovers the standard O(ε) results, but “ε does not need to be small”; the conditions are “stronger than the local exponential stability requirements”.
• Three time scales (Proposition 1, Lemma 3, Theorem 4). With ν₁ = ε₁, ν₂ = ε₂/ε₁ and q = (z, w), if the stacked Jacobian [∂g/∂q; ∂h/∂q] has µ_∞ ≤ −λ, then (10) is partially contracting in q at rate λ min(1, 1/ν₂)/ν₁, and the errors are O(ε₁) + O(ε₂). The µ_∞ argument is a row-wise check and is right. The O(ε₂) term arises because C₁, C₂ = O(ε₂/ε₁) when ν₂ > 1.
• Modularity (Definition 2, Theorem 5). |y(t) − y_is(t)| ≤ K₀e^{−λG₁t} + K₁/G₁ for G₁ > G* = 2λ_x/min(λ_y, λ_z), even when the load’s rate G₂ = βG₁ is as large as G₁, because P1 and P2 make the load act on the slow manifold only through C s = 0 at equilibrium.
• The phosphorylation example. u = Z (kinase), y = X*, v = C; λ_x = δ, λ_y = ηδ, α = 0, so C₁ = C₂ = 0; the connected system contracts in µ₁ with Θ_z = diag(1, θ), θ ∈ (1, 1 + a/b). The bound K₁/G₁ = 2 × 10⁻⁶ against a simulated error of 5 × 10⁻⁸, “a factor of about 40”, which the authors trace to a factor of about 40 between J’s eigenvalues.

Claims

id	claim	strength	support
C1	A contracting system disturbed by d stays within χe^{−λt}|Δ(0)| + χd/λ of an undisturbed trajectory	strong, for constant Θ	Lemma 1; the differential inequality cited to Lohmiller–Slotine and Desoer–Vidyasagar; re-derived here (R1)
C2	The fast state approaches the slow manifold at rate λ_z/ε and stays within εχ_z(d + K)/λ_z of it	strong	Lemma 2; re-derived (R3)
C3	Under partial contraction in x and z, the reduced trajectory is within (7) of the true one for all t ≥ 0	wrong as stated; strong with the reduced system contracting	Theorem 3; counterexample and corrected re-derivation (R4)
C4	The bounds hold at any ε, not only as ε → 0	strong, under the corrected hypotheses	Lemma 2, Theorem 3
C5	With three time scales the errors are O(ε₁) + O(ε₂)	moderate	Theorem 4, from Theorem 3 (so with the same gap) and Proposition 1
C6	Interconnections satisfying P1–P2 are functionally modular for G₁ large, even with G₂ comparable to G₁	moderate	Theorem 5 from Lemmas 4–5; in the example α = 0, so the gap in Theorem 3 does not bite
C7	The bound in the example is conservative by a factor of about 40	weak	one simulation, Fig. 1
C8	The bounds show which parameters to tune (lower p_T/k_d, raise k₂Y_T against k₁X_T), “information [that] could not be obtained by the singular perturbation analysis”	moderate	the form of K₁

Method

Write the slow manifold’s image γ(x(t)) as a disturbed solution of the fast virtual system; apply the robustness lemma to bound the gap; write the slow equation as the reduced system disturbed by f(x, z) − f(x, γ(x)); apply the lemma again and integrate against the fast bound. For several time scales, stack the fast variables and check contraction of the stacked system by µ_∞.

Concepts

• Contracting system: any two trajectories converge exponentially; certified here by µ(∂f/∂x) ≤ −λ in some (constant-weighted) norm.
• Partial contraction: contraction of a virtual system in which the other variables are replaced by arbitrary given functions of time.
• Virtual system: a system that the actual trajectory, and the comparison trajectory, both solve (here εẏ = g(x(t), y, ε)).
• Retroactivity: the back-action of a downstream system on an upstream one through the reactions that connect them.
• Functionally modular interconnection (Definition 2): the connected and isolated trajectories differ by at most K₀e^{−λG₁t} + K₁/G₁.

Re-derivations

The record’s, step by step. Norms |·| are Euclidean unless said otherwise; σ̄(·) and σ(·) are the largest and smallest singular values.

R1. Lemma 1, constant metric, any norm (the paper’s frame). Let µ_Θ(∂f/∂x(x, t)) ≤ −λ for all x, t, with µ_Θ(A) = µ(ΘAΘ⁻¹). Put e = x_p − x and w = Θe. Then ẇ = Θ(f(x_p, t) − f(x, t)) + Θd = A(t)w + Θd(t), where A(t) = ∫₀¹ Θ ∂f/∂x(x + se, t) Θ⁻¹ ds (mean-value theorem along the segment) and d(t) = d(x_p(t), t).

1. µ is subadditive and positively homogeneous, so it is convex; the integral is a limit of Riemann sums, so µ(A(t)) ≤ ∫₀¹ µ_Θ(∂f/∂x) ds ≤ −λ.
2. For h > 0, w(t + h) = (I + hA)w(t) + hΘd(t) + o(h), so |w(t + h)| ≤ |I + hA||w(t)| + h|Θd(t)| + o(h). Subtract |w(t)|, divide by h and let h → 0⁺: D⁺|w| ≤ µ(A)|w| + |Θd| ≤ −λ|w| + |Θd|.
3. Comparison lemma: |w(t)| ≤ e^{−λt}|w(0)| + ∫₀ᵗ e^{−λ(t−τ)}|Θd(τ)| dτ.
4. |e| ≤ |Θ⁻¹||w|, |w(0)| ≤ |Θ||e(0)| and |Θd| ≤ |Θ||d|. With χ = |Θ||Θ⁻¹|: |e(t)| ≤ χe^{−λt}|e(0)| + χ∫₀ᵗ e^{−λ(t−τ)}|d(τ)| dτ ≤ χe^{−λt}|e(0)| + χ sup_{τ≤t}|d(τ)|(1 − e

This is the paper’s bound, in a slightly stronger form: only the disturbance along the perturbed trajectory, up to time t, enters. That is the form Bousquet and Slotine’s Lemma 3 uses, with running maxima.

R2. Lemma 1 for a state- and time-dependent metric (not in this paper). Bousquet and Slotine’s Definition 1 allows Θ(x, t), certified by the generalized Jacobian F = (Θ̇ + Θ ∂f/∂x)Θ⁻¹ with Θ̇ = ∂tΘ + Σᵢ ∂{xᵢ}Θ fᵢ along f, and sym(F) ≼ −λI. Assume f and Θ are C¹ and 0 < θ ≤ σ(Θ(x, t)) ≤ σ̄(Θ(x, t)) ≤ Θ̄ for all x, t.

1. For a C¹ curve c: [0, 1] → ℝⁿ define its length at time t, ℓ_t(c) = ∫₀¹ |Θ(c(s), t) c′(s)| ds, and D(t) = inf ℓ_t(c) over curves from x(t) to x_p(t).
2. Flowing a curve by f shrinks its length. Let φ_{t,h} be f’s flow from t to t + h. For fixed s, v(τ) = ∂s φ{t,τ−t}(c(s)) satisfies v̇ = (∂f/∂x)v, so δz = Θv satisfies δż = Θ̇v + Θ(∂f/∂x)v = Fδz, and d|δz|²/dτ = 2δzᵀFδz ≤ −2λ|δz|². So |δz(t + h)| ≤ e^{−λh}|δz(t)|, and integrating over s, ℓ_{t+h}(φ_{t,h}∘c) ≤ e^{−λh}ℓ_t(c). Θ̇ is taken along f, as the certificate requires, because the curve is moved by f alone.
3. The perturbed endpoint moves off the flowed curve by at most h|d|. x_p(t + h) − φ_{t,h}(x_p(t)) = h d(t) + o(h) for d continuous in t. The straight segment between them has length at time t + h of h|Θ(x_p(t), t)d(t)| + o(h).
4. Given η > 0, take c with ℓ_t(c) ≤ D(t) + η. The flowed curve followed by the segment joins x(t + h) to x_p(t + h), so D(t + h) ≤ e^{−λh}(D(t) + η) + h|Θ d| + o(h). Since η is arbitrary, D⁺D(t) ≤ −λD(t) + |Θ(x_p(t), t)d(t)|.
5. Comparison lemma: D(t) ≤ e^{−λt}D(0) + ∫₀ᵗ e^{−λ(t−τ)}|Θd(τ)| dτ.
6. For any curve, |x_p − x| ≤ ∫|c′| ds ≤ ℓ_t(c)/θ, so |x_p(t) − x(t)| ≤ D(t)/θ. The straight segment gives D(0) ≤ Θ̄|x_p(0) − x(0)|, and |Θd| ≤ Θ̄|d|. With χ = Θ̄/θ: |x_p(t) − x(t)| ≤ χe^{−λt}|x_p(0) − x(0)| + χ∫₀ᵗ e^{−λ(t−τ)}|d(τ)| dτ.

Two points the sketches leave implicit. First, χ must be Θ̄/θ, the ratio of the largest singular value anywhere to the smallest anywhere. The supremum of the pointwise condition numbers can be smaller and is not enough. Second, for partial contraction (f depending on another variable’s path y(t)) with Θ depending on y, Θ̇ includes ∂_yΘ·ẏ along the actual path. The certificate must hold for every admissible path, the actual one included, or Θ must not depend on y.

R3. Lemma 2. By Theorem 2 and the definition of γ, g(x, γ(x), 0) = 0. Let y_d(t) = γ(x(t)). Then εẏ_d = ε ∂γ/∂x f(x, z, t) = g(x(t), y_d, ε) + [ε ∂γ/∂x f + g(x, y_d, 0) − g(x, y_d, ε)]. The bracket has norm at most εd + Kε. z(t) solves the undisturbed system εẏ = g(x(t), y, ε), which is contracting at rate λ_z/ε in Θ_z. Divide by ε: the disturbance on ẏ is at most d + K. R1 gives |z(t) − γ(x(t))| ≤ χ_z e^{−(λ_z/ε)t}|z(0) − γ(x(0))| + χ_z(d + K)/(λ_z/ε), which is (5).

R4. Theorem 3, and why it needs the reduced system contracting. Write ẋ = f(x, γ(x), t) + Δ₂(t), with Δ₂ = f(x, z, t) − f(x, γ(x), t) and |Δ₂| ≤ α|z − γ(x)|. The comparison trajectory x_γ solves the reduced system ẋ = f(x, γ(x), t). The step “R = |Θ_x(x − x_γ)|, D⁺R + λ_x R ≤ |Θ_x Δ₂|” is R1 applied to the reduced system, so it needs µ_{Θ_x}(∂f/∂x + ∂f/∂z ∂γ/∂x) ≤ −λ_x. Partial contraction in x bounds only µ(∂f/∂x). Granted the corrected hypothesis:

1. D⁺R ≤ −λ_x R + |Θ_x|α(χ_z e^{−(λ_z/ε)t}E₀ + εχ_z(d + K)/λ_z), with E₀ = |z(0) − γ(x(0))|, by R3.
2. ∫₀ᵗ e^{−λ_x(t−s)}e^{−(λ_z/ε)s} ds = ε(e^{−λ_x t} − e^{−(λ_z/ε)t})/(λ_z − ελ_x), valid for either sign of λ_z − ελ_x and by continuity at 0. ∫₀ᵗ e^{−λ_x(t−s)} ds = (1 − e^{−λ_x t})/λ_x.
3. R(t) ≤ e^{−λ_x t}R(0) + |Θ_x|αχ_z E₀ ε(e^{−λ_x t} − e^{−(λ_z/ε)t})/(λ_z − ελ_x) + |Θ_x|αχ_z ε(d + K)(1 − e^{−λ_x t})/(λ_z λ_x).
4. Multiply by |Θ_x⁻¹|, use R(0) ≤ |Θ_x||x(0) − x_γ(0)|: (7) with the stated C₁ and C₂. (8) follows from |z − γ(x_γ)| ≤ |z − γ(x)| + α_γ|x − x_γ|.

The CDC version’s proof is the same and makes the singular values explicit. Counterexample to the theorem as stated: ẋ = −x + 2 tanh z, εż = −z + 2 tanh x. Partial contraction in x and in z at rates 1 (Θ = I, χ = 1); K = 0; α = 2; γ(x) = 2 tanh x, α_γ = 2; |∂γ/∂x f| = |2 sech²x (−x + 2 tanh z)| ≤ 4.895 for all x, z. The reduced system ẋ = −x + 2 tanh(2 tanh x) has slope 3 at 0 and stable equilibria at ±1.915. With x(0) = x_γ(0) = 0.01 and z(0) = −1, x(20) = −1.915 and x_γ(20) = +1.915 at ε = 0.1 and at ε = 0.01, an error of 3.83, while (7) gives at most 0.98 and 0.098. The CDC version’s introduction assumes “the reduced system and the fast system are each partially contracting”, which is the corrected hypothesis.

R5. What this gives THEORY-771. That entry lacks Lemma 1 in a state- and time-dependent metric and the passage from an arbitrary fast initial state to O(ε) (Bousquet and Slotine’s Lemma 4). R2 supplies the first. For the second, Lemma 4 is not needed under Hypothesis 2: R2’s convolution form gives the whole of Theorem 1 directly. In Bousquet and Slotine’s notation (fast time; x̃ fast, ỹ = y − ȳ; |ỹ(0)| = |Δ|; |ȳ| ≤ M̄; ε_c = β_f/[χ_f(α_f,x + χ_g α_f,y α_g,x/β_g)]; ε < ε_c):

1. Fast line. x̃ ≡ 0 solves dx̃/dt = f̃(x̃, y(t), u) because f̃(0, y, u) = 0, and f̃ is contracting in x̃ at rate β_f for the actual path y(t). With the disturbance εδf, R2 gives |x̃(t)| ≤ χ_f e^{−β_f t}|x̃₀| + εχ_f∫₀ᵗ e^{−β_f(t−τ)}|δf(τ)| dτ. (A)
2. Slow line. ȳ solves dy/dt = εḡ(y, u(εt)), contracting at rate εβ_g in fast time (if Θ_g depends on y and on slow time εt only, its derivative along εḡ is ε times its slow-time derivative, so the rate scales by ε); y solves it with disturbance εδg. R2 gives |ỹ(t)| ≤ χ_g e^{−εβ_g t}|Δ| + εχ_g∫₀ᵗ e^{−εβ_g(t−τ)}|δg(τ)| dτ. (B)
3. Small gain on running maxima X(t) = max_{τ≤t}|x̃|, Y(t) = max_{τ≤t}|ỹ|. With |δf| ≤ d_f + α_f,x X + α_f,y(M̄ + Y) and |δg| ≤ d_g + α_g,x X, (A) and (B) give X ≤ χ_f|x̃₀| + (εχ_f/β_f)(d_f + α_f,x X + α_f,y(M̄ + Y)) and Y ≤ χ_g|Δ| + (χ_g/β_g)(d_g + α_g,x X). Substituting, X ≤ X* = [χ_f|x̃₀| + (εχ_f/β_f)(d_f + α_f,y(M̄ + χ_g|Δ| + χ_g d_g/β_g))]/(1 − ε/ε_c), Y ≤ Y* = χ_g|Δ| + (χ_g/β_g)(d_g + α_g,x X*). This is Bousquet and Slotine’s (11), and it holds for any x̃₀, not only x̃₀ = O(ε). X(t) is finite on every interval where the solution exists, so the bound also rules out finite-time blow-up, and the solution is global.
4. Fast decay. Let D_f = d_f + α_f,x X* + α_f,y(M̄ + Y*). By (A), |x̃(t)| ≤ χ_f|x̃₀|e^{−β_f t} + εχ_f D_f/β_f, so for t ≥ log(1/ε)/β_f, |x̃(t)| ≤ ε(χ_f|x̃₀| + χ_f D_f/β_f). That is Lemma 4’s conclusion on x̃, with its constant written out. D_f stays bounded as ε → 0, for example for ε ≤ ε_c/2.
5. Slow error. Put step 4’s bound into |δg| and (B). Using ε∫₀ᵗ e^{−εβ_g(t−τ)} dτ ≤ 1/β_g and ∫₀ᵗ e^{−εβ_g(t−τ)}e^{−β_f τ} dτ ≤ 1/β_f: |ỹ(t)| ≤ χ_g|Δ|e^{−εβ_g t} + χ_g d_g/β_g + εχ_g α_g,x χ_f(D_f/(β_f β_g) + |x̃₀|/β_f). (C) With d_g = 0, |ỹ(t)| = O(ε) for t ≥ log(1/ε)/(εβ_g). The transient is max(log(1/ε)/β_f, log(1/ε)/(εβ_g)), within the (1/β_f + 1/(εβ_g)) log(1/ε) of Theorem 1. With d_g ≠ 0 the floor χ_g d_g/β_g is O(1), as NOTE-1025 found.

So, given R2, every step of THEORY-771’s statement is proved, with constants, and Lemma 4, its loosest step, is bypassed rather than repaired. R2 is the record’s derivation. The paper supplies the constant-metric case (R1).

Checks (scripts in the session scratchpad).

• R1, constant metric: A = [[−1, 4], [0, −2]] is not contracting in the identity metric (µ₂(A) = 0.56). In the Lyapunov metric it contracts at λ = 0.263 with χ = 3.23. Under a bounded, state- and time-dependent disturbance (|d| ≤ 0.7), the error stayed at most 0.42 of the bound over 50 random initial pairs.
• R5: a 2 + 1 linear system in Bousquet and Slotine’s form, with a non-normal fast block (β_f = 0.445, χ_f = 2.15), β_g = 0.5, α_f,x = 0.45, α_f,y = 0.94, α_g,x = 1.08, ε_c = 0.083, d_f = d_g = 0, a large initial fast state |x̃₀| = 5 and Δ = 0.5. At ε = 0.05, 0.2, 0.5 and 0.9 ε_c, sup|x̃| = 5 ≤ X* and sup|ỹ| = 0.5 ≤ Y*. The largest ratio of |x̃| to bound (A) was 0.04 to 0.44, and of |ỹ| to bound (C) 0.002 to 0.26. After the transient, sup|x̃|/ε ≤ 0.034 and sup|ỹ|/ε ≤ 0.48.
• R4: the counterexample above. The corrected theorem on QUESTION-222’s example (ẋ = −2x + y, ẏ = d(x − y), ε = 1/d, γ(x) = x, reduced rate 1, with d taken along the trajectory): the slow error was 41 to 74% of bound (7) at d = 5, 20, 100 and 500, and sup|x − x_γ|/ε rose from 1.65 to 2.95, first order. The fast bound (5) held after t = 5ε.

Connections

It applies Lohmiller and Slotine (1998) on contraction, Wang and Slotine (2005) on partial contraction and Slotine (2003) on robustness, to the singular-perturbation setting of Tikhonov (1952), Kokotović, Khalil and O’Reilly and Khalil’s textbook. It cites Simon (1962), LIT-905, as the motivation for multiple time scales as modularity. Its application builds on Del Vecchio, Ninfa and Sontag (2008) and Jayanthi and Del Vecchio (2010) on retroactivity. Bousquet and Slotine (2015), LIT-1292, build on it. They need its Lemma 1 for a disturbance that grows with the state (their Remark 3), and they close a feedback loop by small gain where this paper has a cascade.

Bearing on the record

• THEORY-771. Its promote_when asks for a step-by-step re-derivation of two steps, or a reading of a published proof of both, “such as Del Vecchio and Slotine”. The paper proves the first step only for constant metrics (R1). It does not contain the second: its Lemma 2 is the analogous fast bound in a different setting, with a globally bounded manifold drift and no small-gain loop. The re-derivations here meet the first branch of the condition. R2 proves Lemma 1 in a state- and time-dependent metric, with the convolution form Lemma 3 uses. R5 proves the passage to O(ε) from any initial fast state, with constants, and closes Theorem 1 without Lemma 4. Recommended: promote THEORY-771 under ADR-047 v2, with R2 and R5 copied into the decision file, and with three precisions added to its text: χ_f and χ_g are Θ̄/θ, global over state and time; for a metric depending on y, the partial-contraction certificate must hold along every admissible path of y; and the transient is max(log(1/ε)/β_f, log(1/ε)/(εβ_g)) or less, which its title’s sum already bounds. Its existing corrections (ε_c inverted in Remark 8, d_g = 0) stand, and R5 step 5 confirms the second.
• THEORY-773 is filed from this reading: Lemma 2 and Theorem 3, with the reduced system assumed contracting. Active: every step is re-derived (R1, R3, R4).
• QUESTION-221. Nothing new on dependence: deterministic, trajectory error only. What it adds is the cascade form. When the fast state’s disturbance does not grow with the slow state, no small-gain ceiling ε_c is needed, and the O(ε) trajectory error holds at every ε. That matters for a future stochastic version, where the analogue of d is the noise-driven motion of the manifold.
• THEORY-661. A second source, beside THEORY-771, for a uniform first-order rate of the reduction, in a cascade setting. Recommended: if THEORY-661’s “What could have come out otherwise” is revised, cite THEORY-773 alongside THEORY-771. Neither is Simon and Ando’s theorem.
• CLAIM-386. R5 steps 3–4 are the estimate its post-transient orders rested on, now with the general-metric lemma behind them. Lemma 2 is the same estimate in the cascade form. Recommended: the note of 2026-10-20 on CLAIM-386 can cite R5 (or THEORY-771, once promoted) as proved, not Proposed. The claim itself is unchanged.
• QUESTION-222. A partial answer, filed as THEORY-773’s answers:. When the eliminated variable is fast and contracting, the quasi-steady-state reduction is a Markovian model of the coarse variable with prediction error O(ε) for all time, constants explicit: the augmentation needed is none. On A5’s example with c = κd, the kernel bκde^{−dt} → bκδ(t), and the memoryless model’s error is bounded by O(1/d) (checked). The memory term matters only in the initial layer, through C₁ and the e^{−dt}y(0) term. The question stays open where it is hard: slow or noisy eliminated variables, and history-dependent levels. Remark-level reading: singular perturbation is Markovian reduction by time-scale separation, and contraction of the reduced system is what keeps the memoryless error from accumulating.
• THEORY-257. No bearing beyond context: nothing information-theoretic.

No instruction for machine-learning practice; nothing for the anthology.

Limitations

• Theorem 3’s hypothesis is wrong in both versions read (R4). Theorem 4 and Lemmas 4–5 apply Theorem 3, so they inherit the gap. Lemma 4 and the example have α = 0, where it does no harm.
• Constant metrics only. State- and time-dependent metrics are mentioned and not used. R2 extends Lemma 1, and is the record’s.
• The bound d is global (“∀x, ∀z, ∀t”) and so cannot hold for linear f with linear γ. The proofs use it along the trajectory, and the example computes it so. That makes d trajectory-dependent and the bound a posteriori unless d is bounded a priori.
• Lemma 1’s differential inequality is cited, not shown, and its statement needs a global bound on d where the proof gives the stronger running-supremum form.
• Misprints in the revision: λ_z in the example is printed as a max of the two column rates where it is their min (µ₁ is the max of the column sums, so the rate is the min); the text refers to “Theorem 7”, which does not exist (Theorem 5 is meant); and “system (16) is already in the form (15)” is printed twice.
• One example, one simulation per regime, figures with no error metric shown.
• Deterministic. No noise, no dependence measure.
• The version of record was not read; the revision of 22 December 2011 was.

Open questions

• With noise on the fast variable, does a Lemma 2 analogue bound the mean square distance from the manifold by O(ε) times the noise intensity, and the slow error accordingly? That is the bridge to QUESTION-221.
• Is there a version of Theorem 3 between partial contraction in x (not enough) and contraction of the reduced system (enough), for instance contraction of the full system in a metric weighted by ε, as the CDC version’s Theorem 3 gives?
• How conservative are the constants in general? Here by 1.4 to 2.4 times in a linear check and by 40 in the paper’s example.