Ordered intricacy of Shilnikov saddle-focus homoclinics in symmetric systems

The paper rigorously derives the 1D return map and proves the double-homoclinic bifurcation equation |μ| = B0 R^{1−ν0} |μ|^{ν0} sin(Ω0 ln|μ| + φ2), the ∩-shaped geometry, and the universal e^{-2π/Ω0} scaling of both widths and gaps (via Theorem 1 and its construction from the local–global map) . For higher/mixed pulses it gives precise iterative return-map formulations and qualitative embedding/bridging structure, but it stops short of proving a general e^{-2π/Ω0} scaling theorem near each (l−1)-pulse anchor (instead providing recursive constructions and qualitative diagrams) . The model solution reproduces the l=2 result and geometry correctly, but it contains some unjustified steps (e.g., a first-hit estimate z_u = ±|μ| + O(|μ|^{ν0}) and an incorrect ‘higher-order’ claim when 0<ν0<1; the separatrix’s first hit is z0=μ in the global map for r=0) and extends to a general-l scaling law without a fully rigorous justification of the needed nondegeneracy and uniform remainder control .