Torus and hyperchaos in 3D Lotka-Volterra map

The paper documents the bifurcation sequence for the 3D Lotka–Volterra map chiefly via numerical continuation, Lyapunov spectra, and eigenvalue plots, e.g., reporting a supercritical Neimark–Sacker at R ≈ 0.887, a saddle–node of invariant circles near R ≈ 0.943, a subcritical NS of a 2-cycle at R ≈ 1.1104, a saddle–node producing a 6-cycle near R ≈ 1.1112, and a final 6-cycle collision at R ≈ 1.1784, all for α = −0.227, β = −0.228, but without rigorous proofs or validated numerics for these global steps . The candidate solution gives a rigorous, fully worked local analysis for step (1) (NS of the fixed point), including an exact crossing parameter and sign of the first Lyapunov coefficient, but also explains why steps (2)–(5) remain nontrivial and likely open (at least not yet published rigorously) as of the cutoff. Therefore, the underlying problem is likely open as of the cutoff, with only step (1) settled rigorously by the model, while the paper remains numerical/empirical on the global claims.