On the Recursive Sequence xn+1=axn−1b+cxnxn−1

Bashir Al-Hdaibat (), Ramadan Sabra, Mahmoud H. DarAssi and Saleem Al-Ashhab
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Bashir Al-Hdaibat: Department of Mathematics, Faculty of Science, The Hashemite University, Zarqa 13133, Jordan
Ramadan Sabra: Department of Marhematics, Faculty of Science, Jazan University, Jazan 45142, Saudi Arabia
Mahmoud H. DarAssi: Department of Basic Sciences, Princess Sumaya University for Technology, Amman 11941, Jordan
Saleem Al-Ashhab: Department of Mathematics, Faculty of Science, Al-Albayt University, Mafraq 25113, Jordan

Mathematics, 2025, vol. 13, issue 5, 1-20

Abstract: In this paper, we investigate the dynamical behaviors of the rational difference equation x n = ( a x n − 1 ) / ( b + c x n x n − 1 ) with arbitrary initial conditions, where a , b , and c are real numbers. A general solution is obtained. The asymptotic stability of the equilibrium points is investigated, using a nonlinear stability criterion combined with basin of attraction analysis and simulation to determine the stability regions of the equilibrium points. The existence of the periodic solutions is discussed. We investigate the codim-1 bifurcations of the equation. We show that the equation exhibits a Neimark–Sacker bifurcation. For this bifurcation, the topological normal form is computed. To confirm our theoretical results, we performed a numerical simulation as well as numerical bifurcation analysis by using the Matlab package MatContM.

Keywords: difference equations; general solution; stability analysis; topological normal forms; bifurcations; MatContM (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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