Shilnikov Chaos — Spiral Homoclinic Orbit

Parameters

Saddle quantity ρ=|σ|/ω: 0.45 σ (spiral decay): -0.45 ω (spiral rate): 1.0 λ (expanding dir): 0.8

ρ = |σ|/ω = -

Shilnikov condition: ρ < 1

Shilnikov (1965): if a 3D system has a homoclinic orbit to a saddle-focus equilibrium with |Re(eigenvalue)| < Im(eigenvalue), then nearby there exist infinitely many periodic orbits — chaos ensues.