Saddle-center and periodic orbit: dynamics near symmetric heteroclinic connection

The paper proves, with explicit normal forms and a reversible Poincaré map near the saddle periodic orbit, that there is a sequence of parameter values µ_n→0 of the same sign for which homoclinic orbits to the saddle–center exist and are of general type; it even derives the asymptotic µ_n = ν^{2n} v+ / a′(0) (ν∈(0,1)), and shows the linearization is not a rotation (hence general position) . By contrast, the candidate solution asserts a sequence |µ_n| ~ C e^{−ρ T_n} (with T_n≈nP) and fixes the transverse phase offset Δφ independent of n; this misses the necessary amplitude–time coupling near γ (the incoming deviation must itself be O(e^{−ρ T_n})), which yields the correct scaling µ_n ~ e^{−2ρ T_n} rather than e^{−ρ T_n}. The paper’s construction via S^n near γ and the equation a(µ)/(v+ + b(µ)) = f_µ^{2n}(a(µ)(v+ + b(µ))) makes this precise and also establishes the “same-sign” property and general type rigorously .