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

Bashir Al-Hdaibat ^{1, 2}

Ramadan Sabra ³

Mahmoud H. DarAssi ⁴

Saleem Al-Ashhab ^{5, 6}

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Department of Mathematics, The Hashemite University, Zarqa 13133, Jordan |

Department of Mathematics, Faculty of Science, The Hashemite University, Zarqa 13133, Jordan |

Department of Marhematics, Faculty of Science, Jazan University, Jazan 45142, Saudi Arabia

⁴

Department of Basic Sciences, Princess Sumaya University for Technology, Amman 11941, Jordan |

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Department of Mathematics, Al-albayt University, Mafraq 25113, Jordan |

⁶

Department of Mathematics, Faculty of Science, Al-Albayt University, Mafraq 25113, Jordan

Publication type: Journal Article

Publication date: 2025-02-28

MDPI

Journal: Mathematics

scimago Q2

SJR: 0.475

CiteScore: 4.0

Impact factor: 2.3

ISSN: 22277390

DOI: 10.3390/math13050823

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Abstract

In this paper, we investigate the dynamical behaviors of the rational difference equation xn=(axn−1)/(b+cxnxn−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.