Global asymptotic stability and trajectory structure rules of high-order nonlinear difference equation

Qianhong Zhang; Liqin Shen; Qianhong Zhang; Liqin Shen

doi:10.3934/math.20241370

AIMS Mathematics

2024, Volume 9, Issue 10: 28256-28272. doi: 10.3934/math.20241370

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Research article

Global asymptotic stability and trajectory structure rules of high-order nonlinear difference equation

Qianhong Zhang ^,,
Liqin Shen

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

Received: 11 July 2024 Revised: 13 September 2024 Accepted: 23 September 2024 Published: 29 September 2024
MSC : 39A10

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.
- global asymptotic stability,
- trajectory structure,
- difference equation,
- nontrivial solution
Citation: Qianhong Zhang, Liqin Shen. Global asymptotic stability and trajectory structure rules of high-order nonlinear difference equation[J]. AIMS Mathematics, 2024, 9(10): 28256-28272. doi: 10.3934/math.20241370

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Abstract

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.

References

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