Title: Dynamical behaviour of a higher order rational recursive sequence

Authors: Abdul Khaliq; Muhammad Shabbir; Lubna Mustafa; Shaheryar Rustam

Addresses: Department of Mathematics, Riphah International University, Lahore, Pakistan ' Department of Mathematics, University of Engineering and Technology, Lahore, Pakistan ' Department of Mathematics, Virtual University of Pakistan, Lahore, Pakistan ' Department of Mathematics, Riphah International University, Lahore, Pakistan

Abstract: This paper examines the dynamical behaviour of a rational recursive sequence of higher order given by x_k+1 = (δ₁₁x_k +δ₁₂x_k-λ + δ₁₃x_k-μ + δ₁₄x_k-ϑ)/(δ₁₅x_k + δ₁₄x_k-λ + δ₁₄x_k-μ + δ₁₄x_k-ϑ), where the initial conditions x_i, i = -n, -n + 1, ..., 0 with n = max{λ, μ, ϑ} are positive real numbers and δ_i ∈ ℝ⁺, i = 11, 12, ..., 18. We derive sufficient conditions for local stability, boundedness, periodicity, and global stability of the system. This study enriches the qualitative theory of rational difference equations by extending stability and periodicity analysis to higher order systems. We examine the existence and absence of 2-cycle periodic solutions under different parameter settings. The analytical results are further supported and illustrated through numerical simulations.

Keywords: rational difference equation; periodic solution; boundedness; stability.

DOI: 10.1504/IJDSDE.2025.151440

International Journal of Dynamical Systems and Differential Equations, 2025 Vol.14 No.6, pp.488 - 505

Received: 28 Feb 2025
Accepted: 04 Oct 2025
Published online: 30 Jan 2026 *

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