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On the Recursive Sequence x n + 1 = a x n − 1 b + c x n x n − 1

Author

Listed:
  • 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)

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.

Suggested Citation

  • Bashir Al-Hdaibat & Ramadan Sabra & Mahmoud H. DarAssi & Saleem Al-Ashhab, 2025. "On the Recursive Sequence x n + 1 = a x n − 1 b + c x n x n − 1," Mathematics, MDPI, vol. 13(5), pages 1-20, February.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:5:p:823-:d:1603041
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    References listed on IDEAS

    as
    1. Ramazan Karatas & Ali Gelişken & Murat Arı, 2024. "A Generalised Difference Equation and Its Dynamics and Solutions," Mathematics, MDPI, vol. 12(22), pages 1-9, November.
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