Closed-form solutions of systems of nonlinear difference equations and their connections to generalized Fibonacci numbers and related sequences

Hassan J. Al Salman; Hassan J. Al Salman

doi:10.3934/math.20251120

AIMS Mathematics

2025, Volume 10, Issue 11: 25307-25328. doi: 10.3934/math.20251120

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Closed-form solutions of systems of nonlinear difference equations and their connections to generalized Fibonacci numbers and related sequences

Hassan J. Al Salman ^,

Department of Mathematics and Statistics, College of Science, King Faisal University, Hofuf 31982, Alahsa, Saudi Arabia

Received: 29 August 2025 Revised: 22 October 2025 Accepted: 29 October 2025 Published: 04 November 2025
MSC : 39A10, 40A05

In this work, we conducted a comprehensive analytical investigation of an extended nonlinear difference system, which can be regarded as a generalization of a classical nonlinear difference equation. The proposed extension introduces additional variables, allowing for the examination of richer and more intricate dynamical behaviors, particularly those arising from nonlinear interactions among the components. By applying suitable variable transformation techniques, the system is reduced to a linear form, thereby facilitating the derivation of explicit and closed-form solutions. Furthermore, we analyzed the presence of periodic solutions and established their connection to the generalized Fibonacci sequence and other related number sequences, which are known to be highly relevant in various scientific and engineering contexts. Our findings reveal that the system exhibits a periodic structure with a fundamental period of 12, indicating a repeating pattern in the variable evolution. In addition, we proposed a broader generalization of the system through nonlinear functional transformations, which preserve the underlying mathematical framework while allowing the modeling of even more complex behaviors. Several illustrative examples were provided to demonstrate the applicability and effectiveness of the theoretical results.
- system of difference equations,
- nonlinear difference equations,
- closed-form solutions,
- generalized Fibonacci sequence,
- periodic solutions
Citation: Hassan J. Al Salman. Closed-form solutions of systems of nonlinear difference equations and their connections to generalized Fibonacci numbers and related sequences[J]. AIMS Mathematics, 2025, 10(11): 25307-25328. doi: 10.3934/math.20251120

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Abstract

In this work, we conducted a comprehensive analytical investigation of an extended nonlinear difference system, which can be regarded as a generalization of a classical nonlinear difference equation. The proposed extension introduces additional variables, allowing for the examination of richer and more intricate dynamical behaviors, particularly those arising from nonlinear interactions among the components. By applying suitable variable transformation techniques, the system is reduced to a linear form, thereby facilitating the derivation of explicit and closed-form solutions. Furthermore, we analyzed the presence of periodic solutions and established their connection to the generalized Fibonacci sequence and other related number sequences, which are known to be highly relevant in various scientific and engineering contexts. Our findings reveal that the system exhibits a periodic structure with a fundamental period of 12, indicating a repeating pattern in the variable evolution. In addition, we proposed a broader generalization of the system through nonlinear functional transformations, which preserve the underlying mathematical framework while allowing the modeling of even more complex behaviors. Several illustrative examples were provided to demonstrate the applicability and effectiveness of the theoretical results.

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