Source
Bousquet and Slotine (2015), “A contraction based, singular perturbation approach to near-decomposability in complex systems”, arXiv:1512.08464, LIT-1292, read in full in NOTE-1025. The statement is its Hypothesis 2, Lemma 3 and Theorem 1 (Appendix). It stands in, on the point of rates, for Simon and Ando (1961, LIT-1127) and Ando and Fisher (1963, LIT-1294), which the record could not obtain.
What was actually shown
Setting (8)–(9). Fast states x̃ (deviations from the quasi-steady manifold) and slow states y obey
dx̃/dt = f̃(x̃, y, u(εt)) + ε δf(x̃, y, u(εt)), dy/dt = ε ḡ(y, u(εt)) + ε δg(x̃, y, u(εt)),
against the reduced system dȳ/dt = ε ḡ(ȳ, u(εt)). Remark 4 shows how to reach this form from dx/dt = f(x, y, u) + εg_x, dy/dt = εg_y by x̃ = x − x̄(y, u), with x̄ the unique root of f = 0; then δg(0, y) = 0 identically.
Hypotheses (Hypothesis 2). f̃ is partially contracting in x̃, with rate β_f and metric condition number χ_f, and f̃(0, y, u) = 0; ḡ is (χ_g, β_g)-contracting; the reduced solution is bounded, |ȳ| ≤ M̄; u is bounded; |δf| ≤ d_f + α_f,x|x̃| + α_f,y|y| and |δg| ≤ d_g + α_g,x|x̃|; and
ε (χ_f/β_f)(α_f,x + (χ_g/β_g) α_f,y α_g,x) < 1.
The estimate (Lemma 3, (10)–(11)). Apply the contraction robustness bound (Lemma 1) to each line, with M_x̃(t) = max_{τ≤t}|x̃(τ)|, ỹ = y − ȳ and |ỹ(0)| = |Δ|:
|x̃(t)| ≤ χ_f|x̃₀| + ε(χ_f/β_f)(d_f + α_f,x M_x̃ + α_f,y(M̄ + M_ỹ)), |ỹ(t)| ≤ χ_g|Δ| + (χ_g/β_g)(d_g + α_g,x M_x̃).
The second line has no ε: the disturbance ε·δg is divided by the slow contraction rate ε·β_g. Substituting the second into the first and taking the supremum (re-derived here, and it matches the paper’s (11)):
M_x̃ ≤ [χ_f|x̃₀| + ε(χ_f/β_f)(d_f + α_f,y(M̄ + χ_g(|Δ| + d_g/β_g)))] / (1 − ε/ε_c).
So x̃₀ = O(ε) gives M_x̃ = O(ε) for all time. Applying Lemma 1 again to the slow line, |ỹ(t)| ≤ χ_g|Δ|e^{−εβ_g t} + (χ_g/β_g)(d_g + α_g,x M_x̃), which is O(ε) after the slow transient only if d_g is zero or O(ε). The paper’s last line omits d_g. The Remark 4 construction gives d_g = 0; the worked example lists d_g = 1, for which the bound on the slow error is O(1).
Theorem 1. From any initial fast state, Lemma 4 brings x̃ to O(ε) by t = log(1/ε)/β_f while y moves by O(ε log(1/ε)); then Lemma 3 applies, and after a further (1/(εβ_g)) log(1/ε) the system stays in an O(ε) neighbourhood of (0, ȳ).
The critical separation. ε_c = 1/[(χ_f/β_f)(α_f,x + (χ_g/β_g) α_f,y α_g,x)]. Remark 8 prints ε < (χ_f/β_f)(…) = ε_c, the reciprocal; the worked example’s ε_c = √2/7 is the reciprocal form evaluated on its constants (with χ = 1: 2(√2/4 + 4·(3/4)(√2/2)) = 7√2/2). Above ε_c the paper’s building example, with one negative conductance, has several equilibria at 2.5ε_c and is often unstable at 5ε_c (Figs. 2–3).
Linear checks (the record’s, NOTE-1025). Three blocks of three units, dx/dt = (A* + εC)x, A* minus symmetric Laplacians, y the block sums, x̃ = (I − VQ)x, identity metric so χ = 1, d_f = d_g = 0. With a symmetric coupling plus a leak (ε_c = 0.369) and a non-passive one (ε_c = 0.235), the bounds held at every ε from 0.02ε_c to 0.9ε_c. sup|x̃|/ε and sup|y − ȳ|/ε stayed bounded as ε fell (first order). The x̃ bound was within a factor of two at small ε. The bound on y − ȳ was loose by 20 to 1,000 times. Of five random non-passive couplings, the full system first lost stability at 2.8, 6.3, 19.5, 19.8 and 47 times ε_c: the ceiling is sufficient and can be conservative by more than an order of magnitude.
What could have come out otherwise: the slow error could have grown with time, as finite-horizon Tikhonov estimates allow. Contraction of the reduced slow system is what makes it uniform.
What this does not say
- Nothing about dynamical dependence. The systems are deterministic. The bounds are on distances between trajectories, not on the information the rest of the system sends into y. With persistent noise the fast states do not shrink to O(ε) and the estimate does not apply as stated. QUESTION-221 asks about dependence and stays open.
- Not that the slow variables are block aggregates. y is whatever coordinate has dy/dt = O(ε), and the partition into modules is given. For a linear system and block sums y = Qx, that needs QA* = 0, every column of each block summing to zero in continuous time: the equal-column-sum condition of CLAIM-386 and THEORY-755. Neither the coordinates nor the blocks are found.
- Not with a pointwise condition number. χ is the largest singular value of the metric anywhere over the smallest anywhere; the supremum of the pointwise condition numbers can be smaller and does not suffice. A metric that depends on y must certify contraction along every admissible path of y (NOTE-1020, R2).
- Not a necessary condition. ε_c guarantees; it does not mark where the approximation fails.
- Not that the reduced system is stable whenever the modules are. ḡ’s contraction is assumed. Systems with a conserved quantity, as Simon’s building and Simon–Ando’s stochastic blocks are, must first be reduced to remove it.
- Not Simon and Ando’s own statement. It is a different proof in a different frame (continuous time, nonlinear, contraction metrics), and the paper does not compare its constants with theirs.
Connections
- THEORY-661 gains the rate its open statement lacked: first order in ε for trajectory error, uniform in time, under contraction.
- CLAIM-386. Its post-transient orders rested on “the standard estimate that post-transient states lie within O(ε) of the slow subspace, which was not proved here.” This is that estimate, in continuous time, under contraction. With it, the closure defect of the slow variables, ε·δg = ε·QCx̃ in the linear case, is O(ε²) after the transient against a drift of O(ε): the second order CLAIM-386 measured with equal column sums.
- CLAIM-390 bounds dependence for a block’s full state, to second order. This bounds trajectory error, to first order. They measure different things and do not conflict.
- QUESTION-218. Remark 6: the micro states act on the macro level through their quasi-steady values x̄(y), “an aggregation in time”, and act only as a group if the coupling is also low rank, which the paper separates from weakness.
- CLAIM-331. Remark 8’s cascade, τ_µ = (ε_c,µ ⋯ ε_c,M) τ_M, is an “expected” estimate, not a theorem.