Shilnikov Chaos — Saddle-Focus Spiral

Shilnikov's theorem (1965): if a 3D flow has a homoclinic orbit connecting a saddle-focus fixed point where ρ/λ < 1 (the Shilnikov condition), then chaotic dynamics — a Smale horseshoe — exists in any neighborhood. The trajectory spirals outward on the unstable 2D manifold, shoots along the 1D stable manifold back to the origin, and repeats with sensitive dependence. Adjust ρ/λ to cross the chaos boundary.