System: ẋ = -εx + ay, ẏ = -γy + x². When ε≫γ, x relaxes rapidly to the center manifold x* = ay/ε. Substituting eliminates the fast variable: ẏ = -γy + a²y²/ε². The yellow trajectory shows this reduced 1D dynamics. The purple curve is the manifold x = ay/ε. Adiabatic elimination underlies laser rate equations, chemical kinetics (quasi-steady state), and singular perturbation theory.