Adiabatic Elimination of Fast Variables
When one variable relaxes much faster than another (ε ≪ 1), the fast variable slaved to the slow — reducing dimensionality to an effective 1D equation on the slow manifold.
System: ẋ = −x + y², ẏ = −y/ε + x
Fast variable y relaxes at rate 1/ε. Slow manifold: y_s = εx (set ẏ=0 → y=εx).
Effective slow equation: ẋ_eff = −x + ε²x². Gold = slow manifold, blue = full trajectory.