ON THE ASYMPTOTIC BEHAVIOUR OF THE SOLUTIONS OF A FIFTH ORDER DIFFERENCE EQUATION

The paper proves a complete trichotomy for the fifth–order rational recurrence x_{n+1} = a x_{n-1} + (b x_{n-1} x_{n-4})/(c x_{n-4} + d x_{n-2}) with positive parameters/initial data, using the ratio y_{n+1}=x_{n+1}/x_{n-1}, the subsequences w_n := c + d y_{3n}, and a linear second‑order reduction to show lim (x_{n+1}/x_{n-1}) = (ρ_+ − c)/d; from the sign of A := (c + d)(1 − a) − b, it deduces convergence to 0 (A>0) or to +∞ (A<0), and it analyzes the A = 0 border via a 5D linearization whose spectrum is {±1} ∪ roots of λ^3 + bd/(c + d)^2, together with an explicit non‑constant family converging to a positive limit when B := bd − (c + d)^2 ≤ 0 (Theorem 2.2 and its proof; linearization and Ki‑iteration) . The candidate solution proves the same classification by a different, clean route: it reduces to the Möbius ratio iteration r_{n+1} = F(r_{n-2}) with F(y) = a + b/(c + d y), shows all three mod‑3 subsequences of r_n converge to the unique fixed point r* of F (no 2‑cycles for F∘F), and then deduces decay/growth of even/odd subsequences of x_n from r* ≶ 1. It recovers the A=0 spectrum (characteristic polynomial (λ^2 − 1)(λ^3 + μ), μ = bd/(c + d)^2) and notes that, under A=0 with all parameters positive, B = bd − (c + d)^2 = −(c + d)(c + d a) < 0, so the “B > 0” instability clause is vacuous in this strictly positive‑parameter setting—consistent with the paper’s setup. Minor issues in the paper (e.g., a likely typo “B > 1” instead of “B > 0,” and a misphrasing that lists complex eigenvalues as ‘real’) do not affect correctness of results .