nucleation

LIT-1295: A Contraction Theory Approach to Singularly Perturbed Systems

2 min read

Properties5
StatusActive — read 2026-10-20 (NOTE-1020) in full, from the authors' revised manuscript of 22 December 2011 on Del Vecchio's MIT pages, with the CDC-ECC 2011 conference version (MIT DSpace) for comparison; worth reading for the contraction robustness lemma with its one-line proof (Lemma 1: a system contracting at rate λ in a constant metric Θ, disturbed by d with |d| ≤ d̄, stays within χe^{−λt}|Δ(0)| + χd̄/λ of an undisturbed trajectory), for an explicit, global, uniform-in-time bound on the distance of the fast state from the slow manifold (Lemma 2: χ_z e^{−λ_z t/ε}|z(0) − γ(x(0))| + εχ_z(d + K)/λ_z), and for an explicit O(ε) bound on the error of the quasi-steady-state (Markovian) reduction for all t ≥ 0 (Theorem 3), extended to three time scales and applied to retroactivity in a phosphorylation cycle. Read it knowing that Theorem 3 as stated is false: it assumes only that f is partially contracting in the slow state x, while its proof uses contraction of the reduced system ẋ = f(x, γ(x), t), and a two-variable example satisfying every stated hypothesis has a bistable reduced system and an O(1) error (NOTE-1020); with the reduced system assumed contracting, which the CDC version's introduction says, the theorem and its proof are correct. Read it also knowing that its metrics are constant (state- and time-dependent metrics are mentioned, not used), that the bound d on |∂γ/∂x · f| is global and cannot hold for a linear system, though the proof needs it only along the trajectory, that the example's contraction rate λ_z is printed as a max where it is a min, and that the copy read is the revised manuscript, not the version of record.
Version2
Tagsmathematics, complex-systems, natural-sciences
Cited as “paper” byNOTE-1020 — Del Vecchio and Slotine — Contraction and singular perturbation
Cited as “source” byTHEORY-773 — In a singularly perturbed system ẋ = f(x, z, t), εż = g(x, z, ε), if g is contracting in z at rate λ_z for every x and ε, the reduced system ẋ = f(x, γ(x), t) on the slow manifold z = γ(x) is contracting at rate λ_x, f is α-Lipschitz in z, |∂γ/∂x · f| ≤ d and g is K-Lipschitz in ε, then for all t ≥ 0 the fast state is within χ_z e^{−λ_z t/ε}|z(0) − γ(x(0))| + εχ_z(d + K)/λ_z of the manifold and the slow state within χ_x|x(0) − x_γ(0)|e^{−λ_x t} + ε[C₁(e^{−λ_x t} − e^{−λ_z t/ε}) + C₂(1 − e^{−λ_x t})] of the reduced trajectory; partial contraction of f in x alone is not enough

Graph View

Del Vecchio, D. & Slotine, J.-J. E. (2013). A Contraction Theory Approach to Singularly Perturbed Systems. IEEE Transactions on Automatic Control 58(3): 752–757. doi:10.1109/TAC.2012.2211444. Conference version: Proc. 50th IEEE CDC-ECC (2011), pp. 5831–5836, doi:10.1109/CDC.2011.6160340, open at MIT DSpace (hdl 1721.1/79864).

Read in full in NOTE-1020, from the authors’ revised manuscript.

Key takeaways

  • Contraction turns singular perturbation’s local, “ε small enough” statements into global bounds with explicit constants, valid at any ε (Lemma 2, Theorem 3).
  • The fast state approaches the slow manifold at rate λ_z/ε and then stays within εχ_z(d + K)/λ_z of it, where d bounds how fast the manifold moves (Lemma 2).
  • The quasi-steady-state reduction, a memoryless (Markovian) model of the slow state, is accurate to O(ε) uniformly in time, but only if the reduced system itself is contracting. Partial contraction of f in x, the hypothesis printed, is not enough (counterexample in NOTE-1020).

Standing in the record

It is the published proof THEORY-771’s promotion condition names for the robustness lemma that Bousquet and Slotine (LIT-1292) build on. It proves that lemma for constant metrics only. The general-metric version that Bousquet and Slotine’s hypotheses need, and the passage of the fast state to O(ε), are re-derived in NOTE-1020, not taken from the paper. THEORY-773 holds the corrected Theorem 3.

Backlinks