Analysis of Trajectory Structure and GAS for a High-Order Nonlinear Difference Equation

Bingyan Li; Qianhong Zhang

doi:10.62762/JMIA.2025.554313

Volume 2, Issue 1, Journal of Mathematics and Interdisciplinary Applications

Volume 2, Issue 1, 2026

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Journal of Mathematics and Interdisciplinary Applications, Volume 2, Issue 1, 2026: 28-35

Open Access | Research Article | 07 March 2026

Analysis of Trajectory Structure and GAS for a High-Order Nonlinear Difference Equation

Bingyan Li 1

Qianhong Zhang 1 *

1 School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China

* Corresponding Author: Qianhong Zhang, [email protected]

DOI: 10.62762/JMIA.2025.554313

ARK: ark:/57805/jmia.2025.554313

Received: 09 November 2025, Accepted: 02 March 2026, Published: 07 March 2026

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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.

Graphical Abstract

Analysis of Trajectory Structure and GAS for a High-Order Nonlinear Difference Equation

Keywords

global asymptotic stability

semi-cycle analysis

trajectory structure

nonlinear difference equation

Data Availability Statement

Data will be made available on request.

Funding

This work was supported in part by the National Natural Science Foundation of China under Grant 12461038; in part by the Guizhou Scientific and Technological Platform Talents under Grant GCC[2022]020-2; in part by the Scientific Research Foundation of Guizhou Provincial Department of Science and Technology under Grant [2022]021 and Grant [2022]026; in part by the Universities Key Laboratory of System Modeling and Data Mining in Guizhou Province under Grant 2023013; in part by the Postgraduate Research Foundation of Guizhou University of Finance and Economics under Grant 2025BAZYSY201.

Conflicts of Interest

The authors declare no conflicts of interest.

AI Use Statement

The authors declare that no generative AI was used in the preparation of this manuscript.

Ethical Approval and Consent to Participate

Not applicable.

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APA Style

Li, B., & Zhang, Q. (2026). Analysis of Trajectory Structure and GAS for a High-Order Nonlinear Difference Equation. Journal of Mathematics and Interdisciplinary Applications, 2(1), 28–35. https://doi.org/10.62762/JMIA.2025.554313

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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  -

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@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}
}

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