Shilnikov Chaos: Spiral Homoclinic Orbit

The Shilnikov theorem predicts chaos when a saddle-focus equilibrium has a homoclinic orbit and the saddle quantity ρ = |Re(λ)| / Re(σ) < 1, where λ is the real eigenvalue and σ is the real part of the complex pair. Trajectories spiral outward through the saddle point, generating a countably infinite set of periodic orbits and chaotic dynamics in its neighborhood.