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On the rule of trajectory structure for a third-order nonlinear difference equation using semi-cycle analysis method

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Abstract

The paper is concerned with the rule of trajectory structure and global asymptotical stability of the positive equilibrium for the following third-order nonlinear difference equation

$$\begin{aligned} w_{m+1}=\frac{w_{m-1}^{\alpha }w_{m-2}+1+c}{w_{m-1}^{\alpha }+w_{m-2}+c},\ \ \ m\in N, \end{aligned}$$

where \(\alpha \in [0,1], c\in [0,\infty )\), the initial values \(w_{j}\in (0,\infty ), j=0,-1,-2.\) By virtue of semi-cycle analysis method, the rules of trajectory structure are tested in prime period 7. The continuous length of positive and negative semi-cycles of any nontrivial solution appears periodically and the rule is \(3^{-},2^{+},1^{-},1^{+}\). Also, the positive equilibrium is globally asymptotically stable. Finally, two examples are given to show effectiveness of our theoretic analysis.

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Acknowledgements

This work was financially supported by National Natural Science Foundation of China (Grant 12461038)), Guizhou Scientific and Technological Platform Talents (GCC[2022] 020-1), Scientific Research Foundation of Guizhou Provincial Department of Science and Technology ([2022]021, [2022]026), Universities Key Laboratory of System Modeling and Data Mining in Guizhou Province (No. 2023013).

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  1. School of Mathematics and Statistics, Guizhou University of Finance and Economics, Huayan Road, Huaxi University City, Guiyang, 550025, Guizhou, China

    Liqin Shen & Qianhong Zhang

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  1. Liqin Shen
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Correspondence to Qianhong Zhang.

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Shen, L., Zhang, Q. On the rule of trajectory structure for a third-order nonlinear difference equation using semi-cycle analysis method. J. Appl. Math. Comput. 71, 453–463 (2025). https://doi.org/10.1007/s12190-024-02247-5

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  • DOI: https://doi.org/10.1007/s12190-024-02247-5

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