On explicit periodic solutions in three-dimensional difference systems

Ahmed Ghezal; Najmeddine Attia; Ahmed Ghezal; Najmeddine Attia

doi:10.3934/math.20251128

AIMS Mathematics

2025, Volume 10, Issue 11: 25469-25488. doi: 10.3934/math.20251128

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On explicit periodic solutions in three-dimensional difference systems

Ahmed Ghezal ¹,
Najmeddine Attia ^{2
,
,}

1.
Department of Mathematics, Abdelhafid Boussouf University of Mila, Mila, Algeria
2.
Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa 31982, Saudi Arabia

Received: 02 August 2025 Revised: 10 October 2025 Accepted: 29 October 2025 Published: 05 November 2025
MSC : Primary 39A10; Secondary 40A05

This paper focuses on the existence and analytical formulation of closed-form solutions for a three-dimensional system of nonlinear difference equations. The proposed system possesses a mathematical architecture that encapsulates complex nonlinear interactions among three mutually dependent variables. Through the application of systematic analytical transformations, the original system was reduced to a set of solvable recurrence relations, thereby allowing the derivation of explicit closed-form expressions with a high degree of analytical precision. Furthermore, numerical examples revealed that even minute perturbations in the system parameters or initial conditions can induce significant variations in oscillatory patterns, highlighting the system's structural sensitivity and rich dynamical diversity. This paper, therefore, constitutes a natural and essential extension of previously studied two-dimensional frameworks toward more sophisticated three-dimensional models, which provide a more faithful representation of interdependent relationships within discrete nonlinear systems.
- system of difference equations,
- nonlinear difference equations,
- closed form,
- periodicity
Citation: Ahmed Ghezal, Najmeddine Attia. On explicit periodic solutions in three-dimensional difference systems[J]. AIMS Mathematics, 2025, 10(11): 25469-25488. doi: 10.3934/math.20251128

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Abstract

This paper focuses on the existence and analytical formulation of closed-form solutions for a three-dimensional system of nonlinear difference equations. The proposed system possesses a mathematical architecture that encapsulates complex nonlinear interactions among three mutually dependent variables. Through the application of systematic analytical transformations, the original system was reduced to a set of solvable recurrence relations, thereby allowing the derivation of explicit closed-form expressions with a high degree of analytical precision. Furthermore, numerical examples revealed that even minute perturbations in the system parameters or initial conditions can induce significant variations in oscillatory patterns, highlighting the system's structural sensitivity and rich dynamical diversity. This paper, therefore, constitutes a natural and essential extension of previously studied two-dimensional frameworks toward more sophisticated three-dimensional models, which provide a more faithful representation of interdependent relationships within discrete nonlinear systems.

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