Shilnikov Chaos — Homoclinic Orbit & Return Map

Shilnikov's theorem: if a 3D system has a homoclinic orbit connecting a saddle-focus with eigenvalues (ρ±iω, λ) where |ρ|<|λ| (Shilnikov condition), then the system is chaotic with infinitely many periodic orbits. The return map near the fixed point has a characteristic horseshoe structure.

3D phase portrait (x-y projection). Trajectory spirals outward near saddle-focus, returns along unstable manifold.

Return map: z_n+1 vs z_n at Poincaré section. Chaos = non-monotone map.

Time series x(t) — irregular oscillations.

ρ (unstable spiral rate): 0.4 ω (spiral frequency): 1.0 λ (return eigenvalue): -1.5

|ρ|/|λ|: —

Shilnikov condition: —

Chaos: —