On the solutions of some systems of rational difference equations

M. T. Alharthi; M. T. Alharthi

doi:10.3934/math.20241463

AIMS Mathematics

2024, Volume 9, Issue 11: 30320-30347. doi: 10.3934/math.20241463

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Research article

On the solutions of some systems of rational difference equations

M. T. Alharthi ^,

Department of Mathematics and Statistics, Faculty of Science, Jeddah University, Jeddah, Saudi Arabia

Received: 04 July 2024 Revised: 27 August 2024 Accepted: 04 October 2024 Published: 24 October 2024
MSC : 39A10

In this paper, we considered some systems of rational difference equations of higher order as follows
$ \begin{eqnarray*} u_{n+1} & = &\frac{v_{n-6}}{1\pm v_{n}u_{n-1}v_{n-2}u_{n-3}v_{n-4}u_{n-5}v_{n-6}}, \\ v_{n+1} & = &\frac{u_{n-6}}{1\pm u_{n}v_{n-1}u_{n-2}v_{n-3}u_{n-4}v_{n-5}u_{n-6}}, \end{eqnarray*} $
where the initial conditions $ u_{0, } $ $ u_{-1}, $ $ u_{-2}, $ $ u_{-3}, $ $ u_{-4}, $ $ u_{-5}, $ $ u_{-6}, $ $ v_{0, } $ $ v_{-1}, $ $ v_{-2}, $ $ v_{-3}, $ $ v_{-4}, $ $ v_{-5} $ and $ v_{-6} $ were arbitrary real numbers. We obtained a closed form of the solutions for each considered system and also some periodic solutions of some systems were found. We presented some numerical examples to explain the obtained theoretical results.
- difference equations,
- periodic solutions,
- stability,
- recursive sequences,
- system of difference equations
Citation: M. T. Alharthi. On the solutions of some systems of rational difference equations[J]. AIMS Mathematics, 2024, 9(11): 30320-30347. doi: 10.3934/math.20241463

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Abstract

In this paper, we considered some systems of rational difference equations of higher order as follows

$ \begin{eqnarray*} u_{n+1} & = &\frac{v_{n-6}}{1\pm v_{n}u_{n-1}v_{n-2}u_{n-3}v_{n-4}u_{n-5}v_{n-6}}, \\ v_{n+1} & = &\frac{u_{n-6}}{1\pm u_{n}v_{n-1}u_{n-2}v_{n-3}u_{n-4}v_{n-5}u_{n-6}}, \end{eqnarray*} $

where the initial conditions $ u_{0, } $ $ u_{-1}, $ $ u_{-2}, $ $ u_{-3}, $ $ u_{-4}, $ $ u_{-5}, $ $ u_{-6}, $ $ v_{0, } $ $ v_{-1}, $ $ v_{-2}, $ $ v_{-3}, $ $ v_{-4}, $ $ v_{-5} $ and $ v_{-6} $ were arbitrary real numbers. We obtained a closed form of the solutions for each considered system and also some periodic solutions of some systems were found. We presented some numerical examples to explain the obtained theoretical results.

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