Trajectory Structure and Global Asymptotic Stability of Fifth‐Order Nonlinear Difference Equations

This study investigates the dynamical properties and cycle length bifurcations of the following fifth‐order nonlinear difference equation: , where the system parameters are non‐negative, denoted as a , b , c ∈ [0, ∞ ), and the initial values are strictly positive, defined as t _i ∈ (0, ∞ ) for i = −4, −3, −2, −1, 0. Utilizing the semicycle analysis technique, we establish that the lengths of successive positive and negative semicycles for any nontrivial solution demonstrate periodic regularities, specifically yielding prime periods of 30 and 2. The precise sequential patterns within a single period manifest as either 2 ⁻ , 5 ⁺ , 1 ⁻ , 1 ⁺ , 2 ⁻ , 1 ⁺ , 3 ⁻ , 2 ⁺ , 5 ⁻ , 1 ⁺ , 1 ⁻ , 2 ⁺ , 1 ⁻ , 3 ⁺ or an alternating 1 ⁻ , 1 ⁺ , 1 ⁻ , 1 ⁺ configuration. Furthermore, the present work rigorously demonstrates the global asymptotic stability of the equation’s positive equilibrium point. To corroborate our analytical findings, numerical simulations consisting of three distinct examples are provided.