Shilnikov Chaos — Saddle-Focus Spiral

Near a saddle-focus equilibrium with a homoclinic orbit, Shilnikov's theorem (1965) guarantees chaos when ρ/λ < 1 (divergence beats contraction). Trajectories spiral out in 2D, return via 1D stable manifold, and repeat — creating a countably infinite set of periodic orbits.

Saddle index ν = ρ/λ 0.60

Spiral frequency ω 1.50

Num Orbits 4

Shilnikov condition

CHAOS

ν = ρ/λ

0.60

Spiral turns/orbit

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Shilnikov: if ρ/λ < 1, the Poincaré return map is horseshoe-type → Smale horseshoe → countably ∞ many unstable periodic orbits