NOTE-1025: Bousquet and Slotine — Contraction and near-decomposability

Contribution

It turns Simon’s near-decomposability into a singular-perturbation statement with explicit constants, for nonlinear systems. Before it, the open statements the record holds (THEORY-661, from LIT-1241) gave the fast–slow separation for ε “small enough”, with no rate. After it, under contraction hypotheses, the error of the reduced slow description is first order in ε for all later time, and there is a computable ceiling ε_c on the coupling below which this is guaranteed.

Key insight

Contraction makes perturbation errors stop accumulating. If the fast modules forget their initial conditions at rate β_f and the reduced slow system forgets its own at rate εβ_g, then a perturbation of size ε in the fast equation costs O(ε/β_f) in the fast states, and the resulting disturbance ε·α_g,x|x̃| in the slow equation, divided by the slow rate εβ_g, costs O(ε) in the slow states, for ever. The loop between the two closes as a small-gain condition, and that condition is ε < ε_c.

Assumptions

• Singular-perturbation form (8), with time already scaled so that the fast dynamics are O(1) and the slow ones O(ε): dx̃/dt = f̃(x̃, y, u(εt)) + εδf, dy/dt = εḡ(y, u(εt)) + εδg. Reaching it needs slow coordinates y with dy/dt = O(ε) and a unique quasi-steady root x̄(y, u) of the fast equation (Remark 4). Both are assumed found.
• Contraction (Definition 1). A metric M = ΘᵀΘ ≻ 0 with Θ̇Θ⁻¹ + Θ∇fΘ⁻¹ ≼ −βI, and the condition number of Θ bounded by χ. Partial contraction: contracting in x for any fixed path y(t).
• Hypothesis 2. f̃ partially (χ_f, β_f)-contracting in x̃ with f̃(0, y, u) = 0; ḡ (χ_g, β_g)-contracting; |ȳ(t)| ≤ M̄; u bounded; |δf| ≤ d_f + α_f,x|x̃| + α_f,y|y|, |δg| ≤ d_g + α_g,x|x̃|; ε(χ_f/β_f)(α_f,x + (χ_g/β_g)α_f,y α_g,x) < 1.
• Nonlinear, deterministic, continuous time. No noise anywhere. Smoothness “sufficient”. Remark 3 says Hypothesis 2 holds for linear systems and finite-gain L-stable perturbations.
• Invariants removed. Systems with a conserved quantity are not contracting. The worked example changes variables to remove total energy first; the paper says some results survive without contraction of ḡ if y is otherwise bounded (Remark 1), and that only the bound on x̃ then survives.

Key results

• Lemma 1 (robustness, bounded disturbance). For f β, χ-contracting, |x(t) − x₀(t)| ≤ χ|x(0) − x₀(0)|e^{−βt} + χ‖d‖/β. Proof sketched along virtual displacements δz = Θδx.
• Lemma 2 (finite-gain disturbance). With |d| ≤ K₀ + K_x|x| and K_x ≤ β/χ, |x − x₀| ≤ |x(0) − x₀(0)|χe^{−(β−χK_x)t} + χ(K₁ + K_x x₀₀)/(β − χK_x). K₁ in the result is K₀ in the hypothesis, and K_x must be strictly below β/χ.
• Proposition 1. Under Hypothesis 1 the hierarchic reduced system (fast x̃ driven by slow y) is contracting, in the metric diag(ε^{1/2}νΘ_x, Θ_y) for ν small, by a Schur complement.
• Lemma 3. Under Hypothesis 2, from (O(ε), y₀ + Δ): 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(χ_f/β_f)(α_f,x + (χ_g/β_g)α_f,y α_g,x)] (11), so x̃ = O(ε) for all t, and |ỹ| ≤ χ_g|Δ|e^{−εβ_g t} + χ_g α_g,x M_x̃/β_g “which becomes O(ε) after the transient”. Re-derived: (11) follows from (10). The final line drops a term χ_g d_g/β_g, which (10) carries and which is O(1) unless d_g = 0.
• Lemma 4. From any x̃₀, at t = log(1/ε)/β_f the state is (O(ε), y₀ + O(εt)). Holds when: ε below some ε₁; the constant K is not fully defined (the proof introduces K₁ and K₂ and then uses K).
• Theorem 1. After a transient (1/β_f + 1/(εβ_g)) log(1/ε), the system stays in an O(ε) neighbourhood of (0, ȳ); the explicit bound is (11).
• Remark 8. “It is required that ε < (χ_f/β_f)(α_f,x + (χ_g/β_g)α_f,y α_g,x) = ε_c.” As printed this is the reciprocal of Hypothesis 2’s condition. The example’s ε_c = √2/7 is the reciprocal form: with β_f = 1/2, β_g = 1/4, α_f,x = √2/4, α_f,y = 3/4, α_g,x = √2/2 and χ = 1, (1/β_f)(α_f,x + α_f,y α_g,x/β_g) = 7√2/2. For a cascade of modules the paper “expects” τ_µ = (ε_c,µ ⋯ ε_c,M)τ_M, not proved.
• The building example (1)–(2), Figs. 1–3. Two floors of two rooms, conductances f inside floors and εg_i across. With f = g₁ = x + ½ sin x and g₂ = −g₁/2 (a reversed air conditioner, so not passive): at ε = 0.5ε_c the reduction holds; at 2.5ε_c there are several equilibria, which the reduction does not foresee; at 5ε_c the system is “often unstable”. The figures were read for their titles and axes only.

Claims

id	claim	strength	support
C1	Under Hypothesis 2, fast states stay O(ε) from the quasi-steady manifold for all time once they start O(ε)	strong	Lemma 3, (10)–(11); (11) re-derived here; rests on Lemma 1, sketched
C2	Under the same, the slow states stay O(ε) from the reduced trajectory after the slow transient	moderate: true when d_g = 0 or O(ε), not as stated	Lemma 3’s last line, which omits d_g; the example sets d_g = 1
C3	From any initial state, the system reaches and stays in an O(ε) neighbourhood of (0, ȳ) after (1/β_f + 1/(εβ_g)) log(1/ε)	moderate	Theorem 1 from Lemmas 3 and 4; Lemma 4’s proof is loose
C4	ε_c gives a quantitative estimate of the timescale separation needed	strong as a sufficient condition; misprinted	Hypothesis 2; Remark 8 inverted; the example’s √2/7 uses the correct form
C5	Above ε_c the reduction can fail qualitatively (multiple equilibria, instability)	moderate	one example, Figs. 2–3; ε_c is not shown to be sharp
C6	Simon’s “aggregate way” means aggregation in time (through x̄(y)), not an ensemble aggregate of microstates	weak	Remarks 6–7, argued, not proved; Remark 7 “can be shown”, not shown
C7	Separations compose multiplicatively down a cascade of modules	weak	Remark 8, “we expect”
C8	The analysis “quantifies and qualifies” existing results on near-decomposability	weak	no comparison with Simon–Ando or Ando–Fisher is made

Method

Write the system in singular-perturbation form (Remark 4); certify contraction of the fast modules (separately per module when f̃ factors, Remark 5, β_f = min β_i, χ_f = max χ_i) and of the reduced slow system in some metric; bound the coupling terms affinely; compute ε_c; apply Theorem 1.

Concepts

• Contracting system: one whose trajectories converge to each other exponentially, certified by a metric in which the symmetric generalized Jacobian is uniformly negative definite. Stronger than stability of an equilibrium; excludes conserved quantities.
• Partial contraction: contraction in x for every fixed input path y(t).
• Finite-gain L-stable perturbation: |d| ≤ K₀ + K_x|x|.
• Microstates / macrostates: the fast x̃ (or x) and the slow y. The macrostates are whatever coordinates are slow, here differences of floor sums, not a fixed aggregation map.
• ε_c: the largest separation of timescales for which Hypothesis 2’s small-gain condition holds.

Connections

It builds on Del Vecchio and Slotine (IEEE TAC 2013) on contraction and singular perturbation, whose Lemma 1 needed δf bounded where this needs it only finite-gain (Remark 3); on Lohmiller and Slotine (1998) for contraction; and on Slotine (2003) for hierarchic contraction. It cites Simon (1962), LIT-905, and neither Simon and Ando (1961, LIT-1127) nor Ando and Fisher (1963, LIT-1294). It does not compare its bounds with theirs or with Tikhonov’s finite-horizon theorem, though uniformity in time is what it adds over the latter.

Bearing on the record

• THEORY-661. It gives the rate THEORY-661 withdrew for want of a source: first order in ε, for trajectory error, uniform in time after a transient of order (1/β_f + 1/(εβ_g)) log(1/ε), under contraction of the modules and of the reduced slow system. That is a different frame from Simon–Ando (nonlinear, continuous time, contraction metrics) and it is not their theorem. Recommended: THEORY-661’s “What could have come out otherwise” and Source sections cite THEORY-771 for the rate, with its hypotheses stated, and keep “no rate” for Simon–Ando’s own statement. Filed as THEORY-771, Proposed: (11) was re-derived, but Lemma 1 and Lemma 4 are sketches.
• QUESTION-221. Not answered. The paper is deterministic and bounds distances between trajectories; it has no noise, no conditional variance and no information measure. Its estimate does not transfer as stated: with persistent noise x̃ is not O(ε). What does transfer is the structure. In the linear case with y the block sums, the slow equation is dy/dt = εQCVy + εQCx̃, so the part of the rest of the system that reaches y beyond y itself is εQCx̃, a term of size ε times the off-manifold state. That is CLAIM-390’s shape (dependence governed by the incoming coupling times the variance of what y cannot see), now for the aggregate. A numerical check of my own, not the paper’s: in a VAR(1) sampled from dx/dt = (A* + εC)x (three blocks of three, A* minus symmetric Laplacians so the block sums are slow, symmetric C with a leak, step 0.5, unit noise), the transfer entropy into the block sums, with 80 lags of their past, was 5.5 × 10⁻⁴, 2.2 × 10⁻⁴, 7.2 × 10⁻⁵, 2.0 × 10⁻⁵ and 5.4 × 10⁻⁶ at ε = 0.2, 0.1, 0.05, 0.025, 0.0125; T/ε² settled from 0.014 to 0.034. That is second order. It is one system, and a proof for the aggregate is still what QUESTION-221 asks for. Recommended: QUESTION-221’s “What the record holds” names THEORY-771 as the trajectory-error rate and says it does not bound dependence.
• CLAIM-386. Its post-transient orders rest on “the standard estimate that post-transient states lie within O(ε) of the slow subspace, which was not proved here.” Lemma 3 is that estimate, under its hypotheses, in continuous time. With it the slow variables’ closure defect ε·δg is ε·α_g,x·O(ε) = O(ε²) after the transient, against a drift of O(ε), which is CLAIM-386’s measured second order in the equal-column-sum case. The continuous-time form also explains why equal column sums matter: block sums are slow coordinates (dy/dt = O(ε)) iff QA* = 0, every column of each block summing to zero. Checked: in the linear examples the defect ε|QCx̃| was 9.6 × 10⁻⁶, 5.6 × 10⁻⁵ and 2.0 × 10⁻⁴ at ε = 0.0074, 0.018 and 0.037 (slope about 2). CLAIM-386 itself is discrete-time and one-step, and is unchanged. Recommended: a note on CLAIM-386 citing THEORY-771 for the estimate, in continuous time.
• CLAIM-390. No conflict: it bounds dependence of a block’s full state at second order in the incoming coupling; this bounds trajectory error at first order. Note that α_f,y (the slow-to-fast gain) and α_g,x (the fast-to-slow gain) enter ε_c only as a product, beside the fast self-coupling α_f,x, so what must be small is the loop through the slow variables, not each direction separately; CLAIM-390 likewise needs only the incoming coupling.
• CLAIM-388 and QUESTION-218. The paper takes the slow coordinates as given and does not test whether a block is needed. Remark 6 is the relevant passage: the micro states influence the macro level and other modules “through their quasi-steady values x̄_i”, and only as a group “if the coupling is not only weak … but also low rank”, a property “quite different from … connection strength”. That is the record’s distinction between a weakly coupled block and a closed collective variable, stated from the other side and not developed. Remark 7, that Simon’s “aggregate way” is aggregation in time, not an ensemble sum, bears on QUESTION-222 (averaging rather than projection) and is asserted, not shown.
• CLAIM-331. Remark 8’s cascade estimate is the formal shape of “higher levels are slower” and is only “expected”.

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

Limitations

• Remark 8’s ε_c is inverted. Hypothesis 2 and the example agree on ε_c = β_f/[χ_f(α_f,x + χ_g α_f,y α_g,x/β_g)]; Remark 8 prints the reciprocal.
• d_g. Lemma 3’s slow-error bound drops χ_g d_g/β_g. The conclusion y − ȳ = O(ε) needs d_g = 0 (which the Remark 4 construction gives, since δg(0, y) = 0) or d_g = O(ε). The worked example’s constants include d_g = 1, for which Lemma 3 as proved gives only an O(1) slow error; a Lipschitz bound on g₁, g₂ would give d_g = 0 there.
• Equation (2). From (1), δ̇_i = −f(2δ_i) + …, not −f(δ_i); the example’s β_f = 1/2 matches the printed −f(δ_i). With the correct argument β_f would be 1, and ε_c larger, so the stated ε_c is still sufficient.
• Proofs are sketches. Lemma 1 cites the virtual-displacement argument briefly; Lemma 4 leaves K undefined; Remark 7 “can be shown” and is not. Lemma 2’s K₁ should be K₀, and its K_x ≤ β/χ should be strict. Hypothesis 1 asks F_y^s “bounded below” where boundedness of its inverse is used.
• Sufficient, not sharp. In five random non-passive linear couplings the full system first lost stability at 2.8 to 47 times ε_c.
• One example, with figures shown for one random set of initial conditions per ε. No other system is worked.
• Deterministic and given coordinates. No noise, no information measure, no search for the partition or the slow variables.
• Unpublished. Five pages of argument, never refereed as far as could be found.

Open questions

• With additive noise, is the transfer entropy into the slow variables O(ε²) under Hypothesis 2, with a constant in β_f, β_g, the gains and the noise covariance? The structure εQCx̃ and the check above suggest it; that is QUESTION-221.
• How conservative is ε_c in general, and is there a matching necessary condition (for instance a loss of contraction of the full system)?
• Does the time-uniform bound extend to systems with conserved quantities without the change of variables, as Simon–Ando’s stochastic blocks need? Remark 1 says only the x̃ bound survives.