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LIT-1292: A contraction based, singular perturbation approach to near-decomposability in complex systems

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Properties6
StatusActive — read 2026-10-20 (NOTE-1025) in full; worth reading for the only rate the record holds for Simon's near-decomposability: with contracting fast modules (rate β_f) and a contracting reduced slow system (rate β_g), and coupling perturbations bounded affinely in the off-manifold state, Lemma 3 and Theorem 1 keep the fast variables within O(ε) of their quasi-steady manifold and the slow variables within O(ε) of the reduced system, uniformly in time after a transient of (1/β_f + 1/(εβ_g)) log(1/ε), for every ε below an explicit critical timescale separation ε_c = β_f / [χ_f(α_f,x + χ_g α_f,y α_g,x / β_g)]; and for Remark 6, that the micro states act on the macro level through their quasi-steady values, "an aggregation in time", not an ensemble aggregate. Read it knowing that it is a five-page preprint, never published in a journal, whose proofs are sketches; that Remark 8 prints ε_c inverted (the worked example's value √2/7 is the reciprocal of the printed expression, and matches Hypothesis 2); that Lemma 3's O(ε) bound on the slow error needs d_g = 0, which the proof's last line assumes silently while the worked example lists d_g = 1; that it is deterministic throughout and bounds trajectory error only, not any information-theoretic dependence; that the slow coordinates and the partition are given, not found; and that ε_c is sufficient and in linear checks conservative by factors of 3 to 50.
Version3
Tagscomplex-systems, mathematics, control-theory
Cited as “grounds” byCLAIM-478 — Scale separation helps make structure and not only marks it: where some degrees of freedom settle fast relative to others, the relation they settle into is a constraint on the system over the window between the two time scales, so what is a realization at one scale works as a constraint at another; and organization can make new separations in turn, as when modes slow near an instability and become order parameters, so time-scale hierarchies and constraints can produce one another
Cited as “paper” byNOTE-1025 — Bousquet and Slotine — Contraction and near-decomposability
Cited as “source” byTHEORY-771 — In a singularly perturbed system, fast states x̃ with dx̃/dt = f̃(x̃, y) + ε·δf and slow states y with dy/dt = ε·ḡ(y) + ε·δg, if f̃ is contracting in x̃ at rate β_f with x̃ = 0 an equilibrium for every y, ḡ is contracting at rate β_g, and |δf| ≤ d_f + α_f,x|x̃| + α_f,y|y|, |δg| ≤ d_g + α_g,x|x̃|, then for every ε below ε_c = β_f / [χ_f(α_f,x + χ_g α_f,y α_g,x / β_g)] the fast states stay within O(ε) of zero and, when d_g = 0, the slow states within O(ε) of the reduced trajectory, uniformly in time after a transient of order (1/β_f + 1/(εβ_g)) log(1/ε)

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Bousquet, G. D. & Slotine, J.-J. E. (2015). A contraction based, singular perturbation approach to near-decomposability in complex systems. arXiv:1512.08464 [math.DS]. Preprint; no journal version found.

Read in full in NOTE-1025.

Key takeaways

  • Simon’s near-decomposability, cast as singular perturbation, has a first-order rate in ε for trajectory error, uniform in time, when both the modules and the reduced slow system are contracting (Theorem 1, Lemma 3).
  • The admissible coupling has an explicit ceiling ε_c built from the contraction rates and the gains of the coupling terms (Hypothesis 2; Remark 8 prints it inverted). It is sufficient, not necessary.
  • The micro level reaches the macro level through quasi-steady values, “an aggregation in time”, not an ensemble sum (Remark 6, Remark 7).
  • Nothing about information flow or dynamical dependence, and nothing about where the slow coordinates or the blocks are.

Standing in the record

The open stand-in for Simon & Ando 1961 (LIT-1127) and Ando & Fisher 1963 (LIT-1294), both Unavailable, on the one point THEORY-661 lacked: a rate in ε. It supplies, under its hypotheses, the post-transient O(ε) estimate on the fast variables that CLAIM-386’s numerical orders rested on unproved. It does not answer QUESTION-221, which asks about dependence, not trajectory error.

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