TY  - JOUR
AU  - Li, Bingyan
AU  - Zhang, Qianhong
PY  - 2026
DA  - 2026/03/07
TI  - Analysis of Trajectory Structure and GAS for a High-Order Nonlinear Difference Equation
JO  - Journal of Mathematics and Interdisciplinary Applications
T2  - Journal of Mathematics and Interdisciplinary Applications
JF  - Journal of Mathematics and Interdisciplinary Applications
VL  - 2
IS  - 1
SP  - 28
EP  - 35
DO  - 10.62762/JMIA.2025.554313
UR  - https://www.icck.org/article/abs/JMIA.2025.554313
KW  - global asymptotic stability
KW  - semi-cycle analysis
KW  - trajectory structure
KW  - nonlinear difference equation
AB  - This article delves into the trajectory structure rules of a specific fifth-order rational difference equation: $$ s_{m+1}=\frac{s_ms_{m-2}s_{m-3}s_{m-4}+s_ms_{m-2}+s_ms_{m-3}+s_{m-2}s_{m-3}+s_{m-4}+a}{s_ms_{m-2}s_{m-3}+s_ms_{m-2}s_{m-4}+s_ms_{m-3}s_{m-4}+s_{m-2}s_{m-3}s_{m-4}+1+a} $$ where the initial conditions satisfy $s_i\in (0,\infty)$, $i=-4,-3,-2,-1,0$, and the parameters $a\in [0,\infty).$ As the initial values vary, the lengths of consecutive positive and negative semi-cycles for non-trivial solutions exhibit a periodic pattern with a prime period of 31. The rule within one period is $1^-, 2^+, 1^-, 1^+, 1^-, 1^+, 2^-, 4^+, 3^-, 2^+, 2^-, 1^+, 5^-, 1^+, 1^-,$ $ 3^+ $. Through the application of this rule,the global asymptotic stability(GAS) of the positive fixed point of the equation is proven. In the end, three instances are utilized to demonstrate the accuracy of the theoretical conclusions.
SN  - 3070-393X
PB  - Institute of Central Computation and Knowledge
LA  - English
ER  - 

@article{Li2026Analysis,
  author = {Bingyan Li and Qianhong Zhang},
  title = {Analysis of Trajectory Structure and GAS for a High-Order Nonlinear Difference Equation},
  journal = {Journal of Mathematics and Interdisciplinary Applications},
  year = {2026},
  volume = {2},
  number = {1},
  pages = {28-35},
  doi = {10.62762/JMIA.2025.554313},
  url = {https://www.icck.org/article/abs/JMIA.2025.554313},
  abstract = {This article delves into the trajectory structure rules of a specific fifth-order rational difference equation: \$\$ s\_{m+1}=\frac{s\_ms\_{m-2}s\_{m-3}s\_{m-4}+s\_ms\_{m-2}+s\_ms\_{m-3}+s\_{m-2}s\_{m-3}+s\_{m-4}+a}{s\_ms\_{m-2}s\_{m-3}+s\_ms\_{m-2}s\_{m-4}+s\_ms\_{m-3}s\_{m-4}+s\_{m-2}s\_{m-3}s\_{m-4}+1+a} \$\$ where the initial conditions satisfy \$s\_i\in (0,\infty)\$, \$i=-4,-3,-2,-1,0\$, and the parameters \$a\in [0,\infty).\$ As the initial values vary, the lengths of consecutive positive and negative semi-cycles for non-trivial solutions exhibit a periodic pattern with a prime period of 31. The rule within one period is \$1^-, 2^+, 1^-, 1^+, 1^-, 1^+, 2^-, 4^+, 3^-, 2^+, 2^-, 1^+, 5^-, 1^+, 1^-,\$ \$ 3^+ \$. Through the application of this rule,the global asymptotic stability(GAS) of the positive fixed point of the equation is proven. In the end, three instances are utilized to demonstrate the accuracy of the theoretical conclusions.},
  keywords = {global asymptotic stability, semi-cycle analysis, trajectory structure, nonlinear difference equation},
  issn = {3070-393X},
  publisher = {Institute of Central Computation and Knowledge}
}