Solutions and local stability of the Jacobsthal system of difference equations

Ahmed Ghezal; Mohamed Balegh; Imane Zemmouri; Ahmed Ghezal; Mohamed Balegh; Imane Zemmouri

doi:10.3934/math.2024175

AIMS Mathematics

2024, Volume 9, Issue 2: 3576-3591. doi: 10.3934/math.2024175

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Research article

Solutions and local stability of the Jacobsthal system of difference equations

1.
Department of Mathematics, Abdelhafid Boussouf University Center of Mila, Algeria
2.
Department of Mathematics, College of Science and Arts, Muhayil, King Khalid University, Abha 61413, Saudi Arabia; mbalegh@kku.edu.sa
3.
Department of Mathematics, University of Annaba, Elhadjar 23, Annaba, Algeria

Received: 16 November 2023 Revised: 27 December 2023 Accepted: 02 January 2024 Published: 09 January 2024
MSC : 39A10, 40A05

We presented a comprehensive theory for deriving closed-form expressions and representations of the general solutions for a specific case of systems involving Riccati difference equations of order $ m+1 $, as discussed in the literature. However, our focus was on coefficients dependent on the Jacobsthal sequence. Importantly, this system of difference equations represents a natural extension of the corresponding one-dimensional difference equation, uniquely characterized by its theoretical solvability in a closed form. Our primary objective was to demonstrate a direct linkage between the solutions of this system and Jacobsthal and Lucas-Jacobsthal numbers. The system's capacity for theoretical solvability in a closed form enhances its distinctiveness and potential applications. To accomplish this, we detailed offer theoretical explanations and proofs, establishing the relationship between the solutions and the Jacobsthal sequence. Subsequently, our exploration addressed key aspects of the Jacobsthal system, placing particular emphasis on the local stability of positive solutions. Additionally, we employed mathematical software to validate the theoretical results of this novel system in our research.
- local stability,
- Jacobsthal numbers,
- general solution,
- system of difference equations,
- Binet formula
Citation: Ahmed Ghezal, Mohamed Balegh, Imane Zemmouri. Solutions and local stability of the Jacobsthal system of difference equations[J]. AIMS Mathematics, 2024, 9(2): 3576-3591. doi: 10.3934/math.2024175

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Abstract

We presented a comprehensive theory for deriving closed-form expressions and representations of the general solutions for a specific case of systems involving Riccati difference equations of order $ m+1 $, as discussed in the literature. However, our focus was on coefficients dependent on the Jacobsthal sequence. Importantly, this system of difference equations represents a natural extension of the corresponding one-dimensional difference equation, uniquely characterized by its theoretical solvability in a closed form. Our primary objective was to demonstrate a direct linkage between the solutions of this system and Jacobsthal and Lucas-Jacobsthal numbers. The system's capacity for theoretical solvability in a closed form enhances its distinctiveness and potential applications. To accomplish this, we detailed offer theoretical explanations and proofs, establishing the relationship between the solutions and the Jacobsthal sequence. Subsequently, our exploration addressed key aspects of the Jacobsthal system, placing particular emphasis on the local stability of positive solutions. Additionally, we employed mathematical software to validate the theoretical results of this novel system in our research.

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