Solutions and local stability of the Jacobsthal system of di ﬀ erence equations

: We presented a comprehensive theory for deriving closed-form expressions and representations of the general solutions for a speciﬁc case of systems involving Riccati di ﬀ erence equations of order m + 1, as discussed in the literature. However, our focus was on coe ﬃ cients dependent on the Jacobsthal sequence. Importantly, this system of di ﬀ erence equations represents a natural extension of the corresponding one-dimensional di ﬀ erence 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 o ﬀ er 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.

An early example of a nonlinear difference equation with a comprehensively derived closed-form solution is the bilinear equation 0. This equation, commonly known as the Riccati difference equation, has been extensively studied in the literature.Over the last two decades, numerous papers have been published on such equations, systems, or systems derived from them associated with number sequences, including Fibonacci, Lucas, Padovan, Tetranacci, Horadam, Pell, Jacobsthal, and Jacobsthal-Lucas sequences (see citations in [29,30,48,[55][56][57][58][59][60][61] and references therein).Fibonacci numbers as well as Jacobsthal sequences remain of great interest for researchers due to their theoretical richness and applications.The Fibonacci sequence in particular is an inexhaustible source of interesting identities, constituting one of the most famous numerical sequences in mathematics.Similar statements apply to Jacobsthal sequences, which scientists have utilized for their fundamental theory and applications (see citations in [62][63][64][65]).In computer science, for instance, Jacobsthal numbers have been employed in conditional instructions to change the flow of program execution.The properties of these numbers were first summarized by Horadam.In this paper, we extend the investigation of the solvability of a bilinear system of difference equations, where the coefficients are dependent on the Jacobsthal sequence.We provide general solutions using the Jacobsthal sequence.To achieve this objective, we examine the ensuing m−dimensional system of difference equations: and the initial values Ω ( j) −v , j ∈ {1, ..., s} , v ∈ {0, ..., m}.Motivated by the above-mentioned investigations on solvability systems of difference equations, the aim of this paper is to provide detailed theoretical explanations for obtaining closed-form formulas and representations for the general solutions of Eq (1.1) and give natural proofs of the results.We also demonstrate the main results on the long-term behavior, particularly local stability, of the solutions to Eq (1.1).

Main results
To tackle the solution of the system (1.1), we need to leverage the insights provided by the following lemmas.
Lemma 2.2.Consider the homogeneous linear difference equation with constant coefficients with initial conditions Ψ 0 , Ψ −1 ∈ R * , then, the solution is given by: where Proof.The difference equation (2.1) is typically solved using the characteristic polynomial: where λ 1 and λ 2 are the roots of this equation, linked to the Lucas-Jacobsthal number.The closed form of the general solution for Eq (2.1) is given by where Ψ 0 , Ψ −1 are initial values such that and we have After some calculations, we obtain: Thus the lemma is proven.
Lemma 2.3.Consider the following rational difference equation, then, the solution is given by: Proof.Using the change of variables Φ n = J k+1 + 2J k Ω n , ∀n ≥ 0, we can express (2.2) as the closed form of the general solution for Eq (2. 3) is given by: Therefore, the lemma is proven.
2.1.On the system (2.4) In this subsection, we examine the following system of 1st-order difference equations: Now, utilizing the last difference equation in (2.4), we obtain: Similarly, we get: , ∀n ≥ 2 and recursively for the above, we can get System (2.4) can be expressed as the following rational difference equation of s th −order: Let Ω n (i) = Ω sn+i , i ∈ {0, 1, ..., s − 1} .For this, we have By Lemma 2.3, the closed form of the general solution of Eq (2.5) is easily obtained.In the following corollary: Corollary 2.1.Let {Ω n , n ≥ 0} be a solution of Eq (2.5), then, for i ∈ {0, 1, ..., s − 1}, where (J k , k ≥ 0) is the Jacobsthal sequence.
Proof.After some preliminary calculations, the characteristic polynomial of Λ m is given by: where According to Rouche's Theorem, all zeros of χ 1 (λ) − χ 2,k (λ) = 0 lie in the unit disc |λ| < 1.Thus, the positive equilibrium point E is locally asymptotically stable.

Numerical simulation results
For the purpose of numerical solutions, we provide illustrative examples to visually validate the theoretical outcomes for each system under consideration in this manuscript.
Illustrative Example 1: The figure below depicts the numerical solution of Eq (2.2) when k = 2 under specified initial condition Ω 0 = −0.09(refer to Figure 1).

Figure 1 .
Figure 1.Graphical representation of the solution of equation when k = 2 under specified initial condition Ω 0 = −0.09.