Shilnikov Chaos — 3D Homoclinic Orbit

A saddle-focus equilibrium with a homoclinic orbit. When ρ/λ < 1 (Shilnikov condition), the return map creates a Smale horseshoe — infinitely many periodic orbits and chaos.

0.15
0.08
2.0
1500

Shilnikov (1965): if ρ/λ < 1, any homoclinic orbit to a saddle-focus generates chaos. The spiral winding near equilibrium combined with the global return map creates countably many periodic orbits. Current ρ/λ ratio: 0.53CHAOTIC