Shilnikov Chaos — Homoclinic Spiral Attractor

Shilnikov (1965): Consider a 3D system with an equilibrium having eigenvalues −ρ ± iω (stable spiral) and λ > 0 (unstable). If a homoclinic orbit returns to the equilibrium and the Shilnikov condition holds: ρ/λ < 1 (λ > ρ), then the system has countably infinite horseshoes near the homoclinic orbit — Shilnikov chaos.

Each return of the trajectory near the saddle-focus involves exponential contraction (rate ρ) and rotation (frequency ω) in the plane, plus exponential expansion (rate λ) along the unstable manifold. The return map is approximately xₙ₊₁ ≈ xₙ^(ρ/λ) · cos(ω·log xₙ) + μ — a logarithmic spiral that folds infinitely, creating the chaotic invariant set.