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QUESTION-287: In a deterministic singularly perturbed system whose eliminated variable is fast and contracting and whose reduced system is contracting, does the memoryless quasi-steady-state reduction predict the coarse variable with an error bounded uniformly in time in terms of the eliminated dynamics?

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Properties5
StatusAnswered
Tagscomplex-systems, mathematics, natural-sciences
Linedistributed-agency
RefinesQUESTION-222 — Beyond the scalar linear Gaussian case and the deterministic case of a fast, contracting eliminated variable, when a coarse variable's reduced dynamics carry memory of the eliminated variables, does it admit a finite-dimensional Markovian augmentation whose prediction error is bounded in terms of the eliminated dynamics, and is a level whose history, not its present state, does the predicting still a level?
Cited as “answers” 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

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The question

QUESTION-222 asks, for a stated class, when a coarse variable whose reduced dynamics carry memory admits a finite Markovian augmentation with prediction error bounded in terms of the eliminated dynamics. The fast, contracting, deterministic class is one where the augmentation can be empty. This question asks whether the bound holds there.

The answer

THEORY-773, Active, from Del Vecchio and Slotine with their Theorem 3’s hypothesis corrected, answers yes. For ẋ = f(x, z, t), εż = g(x, z, ε), with g contracting in z at rate λ_z, the reduced system on z = γ(x) contracting at rate λ_x, f α-Lipschitz in z, |∂γ/∂x · f| ≤ d along the trajectory and g K-Lipschitz in ε:

  • 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 is 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, for all t ≥ 0.

So k = 0 memory coordinates, and the error is O(ε) with constants in λ_z, λ_x, d and K. Partial contraction of f in x alone is not enough. Slow or noisy eliminated variables remain QUESTION-222’s.

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