Shilnikov's theorem (1965): if a 3D system has a homoclinic orbit to a saddle-focus equilibrium with |λ_s| < ρ (spiral rate), chaos emerges via Smale horseshoes. The trajectory spirals outward, returns, spirals again — never repeating.
Shilnikov condition: ρ/λ = 1.50 — chaos when ratio > 1