Dynamics and Bifurcation for a Class of Second-Order Nonlinear Difference Equations
Abstract
We investigate the dynamics of a class of second-order nonlinear iterative maps given by $x_{n+1}= a x_{n-1}^\alpha + b x_n^\alpha$, where the parameters $a, b$, and $\alpha >0$ govern the system's behavior. First, we establish the existence and stability of equilibrium points, showing that at $\alpha = 1$ a transcritical bifurcation occurs, producing an exchange of stability between the two equilibria. Next, via linear stability analysis, we derive necessary and sufficient conditions under which the prime period-two solution exists and is unstable. Furthermore, we identify a pseudo-subcritical flip bifurcation at a critical threshold where the system transitions from saddle-type behavior to repelling behavior, accompanied by the emergence of an unstable prime period-two solution. Our results provide a complete parameter-dependent characterization of the stability and dynamics of the solutions.
Keywords
Nonlinear dynamics, Second-order, Monotone maps, Bifurcations, Stability
References
- [1] A. M. Amleh, E. Camouzis, G. Ladas, On the dynamics of a rational difference equation, part I, Int. J. Difference Equ., 3(1) (2008), 1–35.
- [2] R. Abo-Zeid, On the solutions of a second order difference equation, Mat. Morav., 21(2) (2017), 61–73.
- [3] E. M. Elsayed, B. S. Alofi, Dynamics and solution structures of a nonlinear system of difference equations, Math. Methods Appl. Sci., 49(5) (2026), 3669–3686. https://doi.org/10.1002/mma.8798
- [4] E. M. Elsayed, F. A. Al-Rakhami, N. M. Seyam, Dynamics and general form of the solutions of rational difference equations, J. Math. Comput. Sci., 34(4) (2024), 424–437. https://dx.doi.org/10.22436/jmcs.034.04.08
- [5] U. Krause, The asymptotic behavior of monotone difference equations of higher order, Comput. Math. Appl., 42(3-5) (2001), 647–654. https://doi.org/10.1016/S0898-1221(01)00184-5
- [6] Z. Rafik, A. H. Salas, L. Martinez, On the boundedness of solutions of the difference equation $x_{n+1} = a x_n^{\alpha} + b x_{n-1}^{\alpha}$, $0 < \alpha \le 2$, and its application in medicine, arXiv:2211.16969 (2022).
- [7] S. Stevic, Boundedness and persistence of solutions of a nonlinear difference equation, Demonstr. Math., 36(1) (2003), 99–104.
- [8] S. Stevic, B. Iricanin, W. Kosmala, et al., On a nonlinear second-order difference equation, J. Inequal. Appl., 2022 (2022), Article ID 88. https://doi.org/10.1186/s13660-022-02822-z
- [9] M. R. S. Kulenovic, G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, New York, 2001. https://doi.org/10.1201/9781420035384
- [10] S. Stevic, B. Iricanin, B., Kosmala, W. et al. On a nonlinear second-order difference equation. J Inequal Appl 2022, 88 (2022). https://doi.org/10.1186/s13660-022-02822-z
