Adiabatic Elimination of Fast Variables

Two-timescale system: ẋ = −x/ε + y·x (fast), ẏ = y(1−y) − x² (slow). As ε→0, the fast variable x slaved to y: x* = ε·y·x ≈ 0 (if stable). The reduced 1D flow on the slow manifold approximates full 2D dynamics perfectly when ε ≪ 1.

Phase plane (x, y) + slow manifold

y(t): full (color) vs reduced (white)

ε (timescale ratio)0.15

Initial x₀1.50

Initial y₀0.20

ε = 0.15 | Slow manifold: x* ≈ 0