Shilnikov Chaos: Homoclinic Spiral Attractor
ρ > |λ| saddle-focus condition • spiraling homoclinic orbits • infinite horseshoes
System Parameters
α (saddle divergence)
0.3
ρ (spiral frequency)
0.9
β (contraction)
0.5
Orbit tail length
3000
ẋ = αx − ρy + x(1−r²)
ẏ = ρx + αy + y(1−r²)
ż = −βz + x²
Shilnikov: chaos iff
ρ > |λ| (saddle-focus)
Shilnikov cond: ρ/α =
3.0
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Shilnikov's theorem (1965): if a 3D system has a saddle-focus equilibrium with a homoclinic orbit, and the spiral frequency ρ exceeds the saddle quantity |λ|, then countably infinite horseshoes exist near the orbit — guaranteeing chaos.