dx/dt = −αx(t−τ) — delayed feedback destabilizes fixed points into chaos
The linear DDE dx/dt = −αx(t−τ) is stable when ατ < π/2 ≈ 1.57; above this threshold, oscillations emerge. The Mackey-Glass equation dx/dt = βx(t−τ)/(1+x(t−τ)ⁿ) − γx models blood cell production and was among the first biological systems shown to be chaotic (Mackey & Glass 1977). Delay equations are infinite-dimensional — initial conditions are functions on [−τ, 0].