Closed-form solutions of a new class of three-dimensional nonlinear difference equations

Ahmed Ghezal; Najmeddine Attia; Ahmed Ghezal; Najmeddine Attia

doi:10.3934/math.20251044

AIMS Mathematics

2025, Volume 10, Issue 10: 23518-23533. doi: 10.3934/math.20251044

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Closed-form solutions of a new class of three-dimensional nonlinear difference equations

Ahmed Ghezal ^{1
,
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Najmeddine Attia ^{2
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1.
Department of Mathematics, University of Mila, Mila, Algeria
2.
Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa 31982, Saudi Arabia

Received: 09 August 2025 Revised: 27 September 2025 Accepted: 13 October 2025 Published: 16 October 2025
MSC : Primary 39A10, Secondary 40A05

This paper explored a new three-dimensional nonlinear system of difference equations, capturing intricate dynamic interactions through advanced analytical and computational methods. By employing strategic transformations and analyzing the system's characteristic polynomial roots, we derived exact closed-form solutions for both distinct and repeated root scenarios. Numerical examples were used to validate the theoretical results and demonstrate that even small variations in initial conditions can induce markedly different dynamical behaviors, ranging from stable oscillations to divergent trajectories.
- nonlinear difference equations,
- three-dimensional systems,
- closed-form solutions,
- population dynamics
Citation: Ahmed Ghezal, Najmeddine Attia. Closed-form solutions of a new class of three-dimensional nonlinear difference equations[J]. AIMS Mathematics, 2025, 10(10): 23518-23533. doi: 10.3934/math.20251044

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Abstract

This paper explored a new three-dimensional nonlinear system of difference equations, capturing intricate dynamic interactions through advanced analytical and computational methods. By employing strategic transformations and analyzing the system's characteristic polynomial roots, we derived exact closed-form solutions for both distinct and repeated root scenarios. Numerical examples were used to validate the theoretical results and demonstrate that even small variations in initial conditions can induce markedly different dynamical behaviors, ranging from stable oscillations to divergent trajectories.

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