BIFURCATION OF PERIODIC ORBITS FOR THE N-BODY PROBLEM, FROM A NON GEOMETRICAL FAMILY OF SOLUTIONS.

The paper proves Theorems 3.1 and 3.2 by factoring F(a,b,T)=b F̃(a,b,T), computing the key partials at the pseudo-circular family α(T)=(a0,0,T), and applying the Implicit Function Theorem under the non-resonance mλn+M ≠ p^2(nm+M) (and its even-q variant). The derivations u=Fb and Rat at α(T) match the linearizations f¨+((M+nm)/r0^3)f=0 and y¨+((mλn+M)/r0^3)y=2a0/r0 used by the model, including T0=π/√((M+nm)/r0^3) and T0*=π/(2√((M+nm)/r0^3)), and the invertibility conditions sin(ωr T0)≠0 and sin(ωr T0*)≠0. The model’s boundary maps Φ=(F/b,Rt) and Ψ=(Ft/b,Rt), symmetry, and C^1-regularity arguments reproduce the paper’s proof almost verbatim, differing only in notation and minor algebraic simplifications (e.g., ∂aRt=(2a0)/(r0ωr)sin(ωrT)=2 sin(ωrT) at a0=ωr r0), and explicitly stating quantifiers about neighborhoods. No missing hypotheses beyond standard ODE smooth dependence and staying away from r=0 on a fixed time slab. See the system (2.1), Lemma 2.2, and the Theorem 3.1/3.2 proofs with the key derivative formulas (3.5), (3.7), and (3.8) in the PDF .