Global asymptotic stability and trajectory structure rules of high-order nonlinear difference equation
In this article, global asymptotic stability and trajectory structure of the following high-order nonlinear difference equation $ z_{n+1} = \frac{z_{n-1}z_{n-2}z_{n-4}+z_{n-1}+z_{n-2}+z_{n-4}+b}{z_{n-1}z_{n-2}+z_{n-1}z_{n-4}+z_{n-2}z_{n-4}+1+b}, \quad n\in N, $ are studied, where $ b\in[0, \in...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2024-09-01
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| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241370?viewType=HTML |
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| Summary: | In this article, global asymptotic stability and trajectory structure of the following high-order nonlinear difference equation
$ z_{n+1} = \frac{z_{n-1}z_{n-2}z_{n-4}+z_{n-1}+z_{n-2}+z_{n-4}+b}{z_{n-1}z_{n-2}+z_{n-1}z_{n-4}+z_{n-2}z_{n-4}+1+b}, \quad n\in N, $
are studied, where $ b\in[0, \infty) $ and the initial conditions $ z_{i}\in(0, \infty), i = 0, -1, -2, -3, -4. $ Using the semi-cycle analysis method, in a prime period, a continuous length of positive and negative semi-cycles of any nontrivial solution appears periodically: 2, 3, 4, 6, 12. Moreover, two examples are given to illustrate the effectiveness of theoretic analysis. |
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| ISSN: | 2473-6988 |