Solvability of Three-Dimensional Nonlinear Difference Systems via Transformations and Generalized Fibonacci Recursions

Abstract

This paper presents closed-form solutions for a three-dimensional system of nonlinear difference equations with variable coefficients. The approach employs functional transformations and leverages generalized Fibonacci sequences to construct the solutions explicitly. These solutions reveal a profound connection to generalized Fibonacci recursions. The proposed method is based on sophisticated mathematical transformations that reduce the complex nonlinear system to a solvable linear form, followed by the derivation of general solutions through iterative techniques and harmonic analysis. Furthermore, we extend our results to a generalized class of systems by introducing flexible functional transformations, while rigorously maintaining the required regularity conditions. The findings demonstrate the effectiveness of this methodology in addressing a broad class of complex nonlinear systems and open new perspectives for modeling multivariate dynamical phenomena. The analysis further reveals two distinct dynamical regimes—an unbounded oscillatory growth phase and a bounded cyclic equilibrium—arising from the relative magnitude of the variable coefficients, thereby highlighting the method’s capacity to characterize both amplifying and self-regulating behaviors within a unified analytical framework.

1. Introduction

Difference equations and systems of difference equations continue to attract significant and renewed interest, owing to their rich mathematical structure and wide-ranging applications across various disciplines. These equations serve not only as theoretical tools arising from abstract iterative processes in pure mathematics but also emerge naturally in numerous applied fields, including theoretical physics, mathematical biology, economics, time series analysis, and computer science (see, for example, [1,2,3,4,5,6,7]). Their widespread use stems from their capacity to model phenomena characterized by discrete-time evolution or governed by complex recursive patterns.

A prominent research direction in this field is the analytical investigation of the solvability of difference equations, with particular emphasis on the existence of solutions and their stability properties. In recent years, many iterative models have been revisited and analyzed from a structural standpoint, often leading to their transformation into simpler and more tractable forms—such as linear equations with constant coefficients or reducible systems (see, for example, [8,9,10,11,12,13,14,15,16,17]). This line of research not only deepens the theoretical understanding of discrete dynamical systems but also fosters the development of accurate and efficient numerical methods for simulating complex iterative phenomena.

From a theoretical standpoint, nonlinear difference equations and systems can often be transformed into simpler or more tractable forms through suitable variable transformations, commonly referred to as structural equivalence techniques. Among the most frequently studied transformed forms are linear first- and second-order difference equations, equations with constant coefficients, quasi-linear forms, and difference systems exhibiting Riccati-type behavior (see, for example, [18,19,20,21,22]). These transformations not only simplify the derivation of explicit solutions but also enable a deeper investigation of the qualitative properties of solutions, such as periodicity, iterative dynamics, boundedness, vanishing behavior, and asymptotic divergence. For classical examples of difference equations and solvable systems, particularly those derived from the discrete reformulation of well-known differential models, readers may refer to works such as [23,24,25], where key mathematical models have been revisited using early iterative and differential approaches to reflect their 18th-century origin. In this context, considerable attention has been devoted to the study of coupled nonlinear difference systems characterized by the mutual dependence of variables through rational expressions. A notable example is the system examined in [26], where the interaction between two sequences is modeled by cross-rational relations involving shifted terms of the variables:

$$v_{m+1}^{\left(1\right)}={\displaystyle \frac{v_{m-1}^{\left(1\right)}{v}_m^{\left(2\right)}}{v_m^{\left(2\right)}\pm v_{m-2}^{\left(2\right)}}}, v_{m+1}^{\left(2\right)}={\displaystyle \frac{v_{m-1}^{\left(2\right)}{v}_m^{\left(1\right)}}{v_m^{\left(1\right)}\pm v_{m-2}^{\left(1\right)}}}, \mathrm{for} \mathrm{all} m \in {\mathbb{N}}_0.$$

Building on this foundation, the study presented in [27] introduced a parametric family of coupled systems that generalizes the interaction terms by incorporating powers of delayed variables along with parametric coefficients. The resulting system is expressed as follows:

$$v_{m+1}^{\left(1\right)}={\displaystyle \frac{{\left(v_{m-k+1}^{\left(1\right)}\right)}^p{v}_m^{\left(2\right)}}{a^{\left(1\right)}{\left(v_{m-k}^{\left(2\right)}\right)}^p+b^{\left(1\right)}{v}_m^{\left(2\right)}}}, v_{m+1}^{\left(2\right)}={\displaystyle \frac{{\left(v_{m-k+1}^{\left(2\right)}\right)}^p{v}_m^{\left(1\right)}}{a^{\left(2\right)}{\left(v_{m-k}^{\left(1\right)}\right)}^p+b^{\left(2\right)}{v}_m^{\left(1\right)}}}, \mathrm{for} \mathrm{all} m,p\in {\mathbb{N}}_0, k\in \mathbb{N},$$

where the coefficients $a^{\left(i\right)}$ and $b^{\left(i\right)}$ and the initial values $v_{-\tau }^{\left(i\right)}$, for $\tau =0,1,\dots ,k$ and $i=1,2$, are non-zero real numbers. Further advancements are proposed in [28], where a more complex class of nonlinear coupled systems is examined. These systems incorporate variable coefficients and combine nonlinear structures involving multiple shifts and mixed powers:

$$v_{m+1}^{\left(1\right)}={\displaystyle \frac{v_m^{\left(2\right)}{v}_{m-1}^{\left(2\right)}{\left(v_{m-1}^{\left(1\right)}\right)}^p}{v_m^{\left(1\right)}(a_m^{\left(1\right)}{\left(v_{m-2}^{\left(2\right)}\right)}^q+b_m^{\left(1\right)}{v}_m^{\left(2\right)}{v}_{m-1}^{\left(2\right)})}}, v_{m+1}^{\left(2\right)}={\displaystyle \frac{v_m^{\left(1\right)}{v}_{m-1}^{\left(1\right)}{\left(v_{m-1}^{\left(2\right)}\right)}^q}{v_m^{\left(2\right)}(a_m^{\left(2\right)}{\left(v_{m-2}^{\left(1\right)}\right)}^p+b_m^{\left(2\right)}{v}_m^{\left(1\right)}{v}_{m-1}^{\left(1\right)})}}, \mathrm{for} \mathrm{all} m\in {\mathbb{N}}_0,p,q\in \mathbb{N},$$

(1)

where the coefficients ${\left(a_m^{\left(i\right)}\right)}_{m\in {\mathbb{N}}_0}$ and ${\left(b_m^{\left(i\right)}\right)}_{m\in {\mathbb{N}}_0}$ and the initial values $v_{-\tau }^{\left(i\right)}$, $\tau =0,1,2$ and $i=1,2$, are non-zero real numbers. Such systems provide a fertile ground for theoretical exploration, particularly in examining the impact of coefficient variability and multi-order interactions on system dynamics. Motivated by these advancements, the present study seeks to extend this line of research by investigating a three-dimensional nonlinear coupled difference system characterized by cyclic interdependence among three components. In this system, each equation governs the evolution of one component as a function of the other two, incorporating both nonlinear terms and time-dependent coefficients:

$$\begin{array}{c}\hfill v_{m+1}^{\left(1\right)}={\displaystyle \frac{v_m^{\left(2\right)}{v}_{m-1}^{\left(2\right)}{\left(v_{m-1}^{\left(1\right)}\right)}^{r_1}}{v_m^{\left(1\right)}\left(a_m^{\left(1\right)}{\left(v_{m-2}^{\left(2\right)}\right)}^{r_2}+b_m^{\left(1\right)}{v}_m^{\left(2\right)}{v}_{m-1}^{\left(2\right)}\right)}},\\ \hfill v_{m+1}^{\left(2\right)}={\displaystyle \frac{v_m^{\left(3\right)}{v}_{m-1}^{\left(3\right)}{\left(v_{m-1}^{\left(2\right)}\right)}^{r_2}}{v_m^{\left(2\right)}\left(a_m^{\left(2\right)}{\left(v_{m-2}^{\left(3\right)}\right)}^{r_3}+b_m^{\left(2\right)}{v}_m^{\left(3\right)}{v}_{m-1}^{\left(3\right)}\right)}},\\ \hfill v_{m+1}^{\left(3\right)}={\displaystyle \frac{v_m^{\left(1\right)}{v}_{m-1}^{\left(1\right)}{\left(v_{m-1}^{\left(3\right)}\right)}^{r_3}}{v_m^{\left(3\right)}\left(a_m^{\left(3\right)}{\left(v_{m-2}^{\left(1\right)}\right)}^{r_1}+b_m^{\left(3\right)}{v}_m^{\left(1\right)}{v}_{m-1}^{\left(1\right)}\right)}},\end{array}$$

(2)

for all $m\in {\mathbb{N}}_0$, $r_i\in \mathbb{N},$ ${(a_m^{\left(i\right)},b_m^{\left(i\right)})}_{m\in {\mathbb{N}}_0},$ $v_{-\tau }^{\left(i\right)}\in {\mathbb{R}}^{\ast },$ and for $\tau =0,1,2$ and $i=1,2,3.$ The principal objective of this study is to establish explicit closed-form solution formulas for the proposed three-dimensional nonlinear systems and to explore their intrinsic connection with generalized Fibonacci-type recurrence relations of the form:

$$F_{m+2}^{\left(i\right)}=F_{m+1}^{\left(i\right)}+r_i{F}_m^{\left(i\right)}, \mathrm{for} \mathrm{all} m\in {\mathbb{N}}_0, \mathrm{and} i=1,2,3,$$

with $F_0^{\left(i\right)}=F_1^{\left(i\right)}=1, i=1,2,3.$ We also extend this model to a broader class by introducing nonlinear functional transformations $f_i,$$i=1,2,3$ applied to each variable. This leads to a structurally transformed system that captures a wider range of nonlinear interactions and dynamic behaviors, as detailed below:

$$\begin{array}{c}\hfill v_{m+1}^{\left(1\right)}=f_1^{-1}\left({\displaystyle \frac{f_2\left(v_m^{\left(2\right)}\right)f_2\left(v_{m-1}^{\left(2\right)}\right){\left(f_1\left(v_{m-1}^{\left(1\right)}\right)\right)}^{r_1}}{f_1\left(v_m^{\left(1\right)}\right)\left(a_m^{\left(1\right)}{\left(f_2\left(v_{m-2}^{\left(2\right)}\right)\right)}^{r_2}+b_m^{\left(1\right)}{f}_2\left(v_m^{\left(2\right)}\right)f_2\left(v_{m-1}^{\left(2\right)}\right)\right)}}\right),\\ \hfill v_{m+1}^{\left(2\right)}=f_2^{-1}\left({\displaystyle \frac{f_3\left(v_m^{\left(3\right)}\right)f_3\left(v_{m-1}^{\left(3\right)}\right){\left(f_2\left(v_{m-1}^{\left(2\right)}\right)\right)}^{r_2}}{f_2\left(v_m^{\left(2\right)}\right)\left(a_m^{\left(2\right)}{\left(f_3\left(v_{m-2}^{\left(3\right)}\right)\right)}^{r_3}+b_m^{\left(2\right)}{f}_3\left(v_m^{\left(3\right)}\right)f_3\left(v_{m-1}^{\left(3\right)}\right)\right)}}\right),\\ \hfill v_{m+1}^{\left(3\right)}=f_3^{-1}\left({\displaystyle \frac{f_1\left(v_m^{\left(1\right)}\right)f_1\left(v_{m-1}^{\left(1\right)}\right){\left(f_3\left(v_{m-1}^{\left(3\right)}\right)\right)}^{r_3}}{f_3\left(v_m^{\left(3\right)}\right)\left(a_m^{\left(3\right)}{\left(f_1\left(v_{m-2}^{\left(1\right)}\right)\right)}^{r_1}+b_m^{\left(3\right)}{f}_1\left(v_m^{\left(1\right)}\right)f_1\left(v_{m-1}^{\left(1\right)}\right)\right)}}\right),\end{array}$$

(3)

for all $m\in {\mathbb{N}}_0$, where $f_i$: $\Gamma ⟶$$\mathbb{R}$, $i=1,2,3$ are one-to-one continuous functions on $\Gamma \subset$$\mathbb{R}$, and $r_i\in \mathbb{N},i=1,2,3$, $v_{-\tau }^{\left(i\right)}$, $\tau =0,1,2$, and $i=1,2,3$ are real numbers in $\Gamma ,$ and ${(a_m^{\left(i\right)},b_m^{\left(i\right)})}_{m\in {\mathbb{N}}_0}$, $i=1,2,3$ are non-zero real numbers. This generalization not only encompasses the previously studied models as special cases but also provides a flexible framework for analyzing more intricate dynamics in discrete-time multivariable systems. The primary objective of this paper is to derive explicit solution formulas for both systems (2) and (3) while emphasizing the theoretical significance of the structural transformation methodology. Our results thus make a novel contribution to the literature on solvable nonlinear systems.

2. Closed-Form Solutions for the Nonlinear System (2) and Its Generalized Extension (3)

In this section, we conduct a comprehensive analysis of the solutions of the nonlinear difference system (2) and its generalized form (3) by employing linear transformation and iterative techniques. We establish the existence of explicit closed-form solutions for (2) under the stated conditions, providing precise recurrence relations for the new variables introduced during the analysis. Our approach is based on a systematic reformulation and transformation of the original equations into simpler forms that can be explicitly handled. Furthermore, we extend these results to the generalized system (3), which involves one-to-one functions, and demonstrate how this system can be reduced to an equivalent linear form through the application of inverse functions. This generalization not only elucidates the structural properties of the studied system but also paves the way for analyzing a broad class of nonlinear systems of a similar type. The findings presented herein offer both deep theoretical insights and practical tools for addressing analogous systems across various domains of applied mathematics. The significance of these results lies in providing researchers with rigorous mathematical methods to better understand the dynamics of complex systems, particularly in exploring their stability and long-term behavior.

We begin by revisiting a classical principle from the lemma of linear difference equations, which yields an explicit formula for solving linear difference equations.

Lemma 1

([28]). Let ${(a_m,b_m)}_{m\in {\mathbb{N}}_0}$ be a given sequence of real numbers. Consider the linear difference equation defined by

$$v_{m+s}=a_m{v}_m+b_m,m\in {\mathbb{N}}_0, s=2,3.$$

Then, the solution of this equation can be explicitly expressed in closed form as:

$$v_{sm+\tau }=\left\{{\displaystyle \prod _{k=0}^{m-1}}{a}_{sk+\tau }\right\}{v}_{\tau }+{\displaystyle \sum _{l=0}^{m-1}}\left\{{\displaystyle \prod _{k=l+1}^{m-1}}{a}_{sk+\tau }\right\}{b}_{sl+\tau }, m\in {\mathbb{N}}_0,$$

where $\tau =0,1,\dots ,s-1$. The following standard conventions are applied: ${\displaystyle \prod _{k=l}^m}{a}_k=1$ and ${\displaystyle \sum _{k=l}^m}{a}_k=0$ whenever $l>m.$ Furthermore, in the special case where the coefficients are constant, that is, $(a_m,b_m)=(a,b)$ for all $m,$ the closed-form solution simplifies significantly to

$$v_{sm+\tau }=\left\{\begin{array}{c}{a}^m{v}_{\tau }+(a^m-1)b/(a-1) \mathrm{if} a\ne 1\hfill \\ v_{\tau }+mb \mathrm{if} a=1\hfill \end{array},\hspace{1em}m\in {\mathbb{N}}_0, \tau =0,1,\dots ,s-1.\right.$$

Before proceeding to establish the necessary conditions for ensuring the well-definedness of solutions, we emphasize that the nonlinear system (2) exhibits a complex recursive structure, linking its variables through nonlinear relationships. The main challenge lies in preserving the regularity of this recursive structure during the forward iterations of the system. Therefore, verifying that the denominators in the recursive expressions remain nonzero for all values of m is a crucial requirement not only for the existence of a solution but also for guaranteeing its continuity and uniqueness. To this end, we make the following remark.

Remark 1.

The following well-definedness condition

$${\displaystyle \prod _{i=1}^3}{v}_m^{\left(i\right)}\left(a_m^{\left(i\right)}{\left(v_{m-2}^{\left(\right(i+1\left)\mathrm{mod}3\right)}\right)}^{r_{(i+1)\mathrm{mod}3}}+b_m^{\left(i\right)}{v}_m^{\left(\right(i+1\left)\mathrm{mod}3\right)}{v}_{m-1}^{\left(\right(i+1\left)\mathrm{mod}3\right)}\right)\ne 0, m\in {\mathbb{N}}_0,$$

(4)

referred to as condition (4)—is a fundamental prerequisite for ensuring that the iterative structure of the nonlinear system (2) remains well-defined at every step of the iteration. This condition guarantees that all algebraic and functional expressions involved in the iteration process remain nonzero, thereby preventing divisions by zero or singularities arising from undefined operations. Without this restriction, the recursive mechanism may break down or lose its mathematical validity after a certain number of steps, potentially compromising the solution construction process or undermining the existence and uniqueness properties essential to the analysis of recursive systems. Therefore, this is not merely a technical requirement but a critical condition that secures both the theoretical soundness and numerical consistency of the explicit closed-form solution. Moreover, it plays a pivotal role in ensuring that the system can serve as a reliable model for analysis or simulation, particularly when applied to mathematical or physical problems where solution continuity and the integrity of the iterative process over extended periods are indispensable.

In the study of nonlinear systems with recursive structures, the ternary system (2) has attracted particular interest due to its intricate cyclic pattern and the strong interdependence among its variables. In this paper, we aim to establish an exact analytical formula that captures the evolution of the system’s variables over time.

Theorem 1.

Consider the nonlinear system (2), with initial values $v_{-\tau }^{\left(i\right)}\in \mathbb{R}$ for $i=1,2,3$ and $\tau =0,1,2$, such that condition (4) is satisfied. Then the nonlinear system (2) admits an explicit closed-form solution, for all $m\in {\mathbb{N}}_0$, $i=1,2,3,$ and τ$=0,1,2$, we have:

$$\begin{array}{cc}{v}_{6m+2\tau }^{\left(i\right)}\hfill & ={\displaystyle \frac{{\left(v_{-2}^{\left(i\right)}\right)}^{r_i{F}_{6m+2\tau }^{\left(i\right)}}}{{\left(v_{-1}^{\left(i\right)}\right)}^{F_{6m+2\tau +1}^{\left(i\right)}}{\left(V_{3(2m+\tau )-\tau }^{\left(i\right)}\right)}^{F_0^{\left(i\right)}}}}\left\{{\displaystyle \prod _{k=0}^{m-1+\left[{\displaystyle \frac{\tau +1}{2}}\right]}}{\displaystyle \frac{{\left(V_{3\left(2k\right)+1}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau )-1}^{\left(i\right)}}}{{\left(V_{3\left(2k\right)}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau )}^{\left(i\right)}}}}\right\}\hfill \\ & \times \left\{{\displaystyle \prod _{k=0}^{m-1+\left[{\displaystyle \frac{\tau }{2}}\right]}}{\displaystyle \frac{{\left(V_{3(2k+1)}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau -1)-1}^{\left(i\right)}}}{{\left(V_{3\left(2k\right)+2}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau -1)}^{\left(i\right)}}}}\right\}\left\{{\displaystyle \prod _{k=0}^{m-1}}{\displaystyle \frac{{\left(V_{3(2k+1)+2}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau -2)-1}^{\left(i\right)}}}{{\left(V_{3(2k+1)+1}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau -2)}^{\left(i\right)}}}}\right\},\hfill \\ v_{6m+2\tau +1}^{\left(i\right)}\hfill & ={\displaystyle \frac{{\left(v_{-1}^{\left(i\right)}\right)}^{F_{2(3m+\tau +1)}^{\left(i\right)}}}{{\left(v_{-2}^{\left(i\right)}\right)}^{r_i{F}_{2(3m+\tau )+1}^{\left(i\right)}}}}\left\{{\displaystyle \prod _{k=0}^m}{\displaystyle \frac{{\left(V_{3\left(2k\right)}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau )+1}^{\left(i\right)}}}{{\left(V_{3\left(2k\right)+1}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau )}^{\left(i\right)}}}}\right\}\hfill \\ & \times \left\{{\displaystyle \prod _{k=0}^{m-1+\left[{\displaystyle \frac{\tau +1}{2}}\right]}}{\displaystyle \frac{{\left(V_{3\left(2k\right)+2}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau -1)+1}^{\left(i\right)}}}{{\left(V_{3(2k+1)}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau -1)}^{\left(i\right)}}}}\right\}\left\{{\displaystyle \prod _{k=0}^{m-1+\left[{\displaystyle \frac{\tau }{2}}\right]}}{\displaystyle \frac{{\left(V_{3(2k+1)+1}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau -2)+1}^{\left(i\right)}}}{{\left(V_{3(2k+1)+2}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau -2)}^{\left(i\right)}}}}\right\},\hfill \end{array}$$

(5)

where [x] denotes the integer part (floor) of x, the sequences ${\{V_m^{\left(i\right)},i=1,2,3\}}_{m\in {\mathbb{N}}_0}$ for all $m\in {\mathbb{N}}_0$, and τ$=0,1,2$, are given by:

$$\begin{array}{cc}\hfill V_{3m+\tau }^{\left(i\right)}=& \left\{{\displaystyle \prod _{k^{\prime }=0}^{m-1}}\left\{{\displaystyle \prod _{k=0}^2}{a}_{3k^{\prime }+\tau +2-k}^{\left(\right(i+k\left)\mathrm{mod}3\right)}\right\}\right\}{\left(v_{\tau -2}^{\left(i\right)}\right)}^{r_i}/\left(v_{\tau -1}^{\left(i\right)}{v}_{\tau }^{\left(i\right)}\right)\hfill \\ & +{\displaystyle \sum _{l^{\prime }=0}^{m-1}}\left\{{\displaystyle \prod _{k^{\prime }=l^{\prime }+1}^{m-1}}\left\{{\displaystyle \prod _{k=0}^2}{a}_{3k^{\prime }+\tau +2-k}^{\left(\right(i+k\left)\mathrm{mod}3\right)}\right\}\right\}{\displaystyle \sum _{l=0}^2}\left\{{\displaystyle \prod _{k=0}^{l-1}}{a}_{3l^{\prime }+\tau +2-k}^{\left(\right(i+k\left)\mathrm{mod}3\right)}\right\}{b}_{3l^{\prime }+\tau +2-l}^{\left(\right(i+l\left)\mathrm{mod}3\right)}.\end{array}$$

Proof. 

Starting from system (2), we establish the following fundamental relation:

$$\begin{array}{c}\hfill {\displaystyle \frac{v_m^{\left(i\right)}{v}_{m+1}^{\left(i\right)}}{{\left(v_{m-1}^{\left(i\right)}\right)}^{r_i}}}={\displaystyle \frac{v_m^{\left(\right(i+1\left)\mathrm{mod}3\right)}{v}_{m-1}^{\left(\right(i+1\left)\mathrm{mod}3\right)}}{a_m^{\left(i\right)}{\left(v_{m-2}^{\left(\right(i+1\left)\mathrm{mod}3\right)}\right)}^{r_{(i+1)\mathrm{mod}3}}+b_m^{\left(i\right)}{v}_m^{\left(\right(i+1\left)\mathrm{mod}3\right)}{v}_{m-1}^{\left(\right(i+1\left)\mathrm{mod}3\right)}}}, m\in {\mathbb{N}}_0, \mathrm{and} i=1,2,3,\end{array}$$

then

$$\begin{array}{c}\hfill {\displaystyle \frac{{\left(v_{m-1}^{\left(i\right)}\right)}^{r_i}}{v_m^{\left(i\right)}{v}_{m+1}^{\left(i\right)}}}=a_m^{\left(i\right)}{\displaystyle \frac{{\left(v_{m-2}^{\left(\right(i+1\left)\mathrm{mod}3\right)}\right)}^{r_{(i+1)\mathrm{mod}3}}}{v_m^{\left(\right(i+1\left)\mathrm{mod}3\right)}{v}_{m-1}^{\left(\right(i+1\left)\mathrm{mod}3\right)}}}+b_m^{\left(i\right)}, m\in {\mathbb{N}}_0, \mathrm{and} i=1,2,3.\end{array}$$

We now introduce the following new variables: $V_m^{\left(i\right)}={\left(v_{m-2}^{\left(i\right)}\right)}^{r_i}$/$\left(v_{m-1}^{\left(i\right)}{v}_m^{\left(i\right)}\right)$, $m\in {\mathbb{N}}_0$, and $i=1,2,3.$ Thus, system (2) can be equivalently reformulated as a closed-cycle three-dimensional linear system of difference equations in terms of the newly introduced variables:

$$V_{m+1}^{\left(i\right)}=a_m^{\left(i\right)}{V}_m^{\left(\right(i+1\left)\mathrm{mod}3\right)}+b_m^{\left(i\right)}, m\in {\mathbb{N}}_0, \mathrm{and} i=1,2,3.$$

Hence, we obtain an equivalent system of three-dimensional linear difference equations involving newly introduced variables, where each variable evolves according to a non-closed recurrence relation:

$$V_{m+3}^{\left(i\right)}=\left\{{\displaystyle \prod _{k=0}^2}{a}_{m+2-k}^{\left(\right(i+k\left)\mathrm{mod}3\right)}\right\}{V}_m^{\left(i\right)}+{\displaystyle \sum _{l=0}^2}\left\{{\displaystyle \prod _{k=0}^{l-1}}{a}_{m+2-k}^{\left(\right(i+k\left)\mathrm{mod}3\right)}\right\}{b}_{m+2-l}^{\left(\right(i+l\left)\mathrm{mod}3\right)}, m\in {\mathbb{N}}_0, \mathrm{and} i=1,2,3.$$

(6)

Based on Lemma 1, the general solution of this system valid for all $m\in {\mathbb{N}}_0$, and for $\tau =0,1,2$, is given by:

$$\begin{array}{cc}\hfill V_{3m+\tau }^{\left(i\right)}=& \left\{{\displaystyle \prod _{k^{\prime }=0}^{m-1}}\left\{{\displaystyle \prod _{k=0}^2}{a}_{3k^{\prime }+\tau +2-k}^{\left(\right(i+k\left)\mathrm{mod}3\right)}\right\}\right\}{V}_{\tau }^{\left(i\right)}\hfill \\ & +{\displaystyle \sum _{l^{\prime }=0}^{m-1}}\left(\left\{{\displaystyle \prod _{k^{\prime }=l^{\prime }+1}^{m-1}}\left\{{\displaystyle \prod _{k=0}^2}{a}_{3k^{\prime }+\tau +2-k}^{\left(\right(i+k\left)\mathrm{mod}3\right)}\right\}\right\}{\displaystyle \sum _{l=0}^2}\left\{{\displaystyle \prod _{k=0}^{l-1}}{a}_{3l^{\prime }+\tau +2-k}^{\left(\right(i+k\left)\mathrm{mod}3\right)}\right\}{b}_{3l^{\prime }+\tau +2-l}^{\left(\right(i+l\left)\mathrm{mod}3\right)}\right).\end{array}$$

We now return to the original variables, where: $v_m^{\left(i\right)}={\left(v_{m-2}^{\left(i\right)}\right)}^{r_i}/\left(v_{m-1}^{\left(i\right)}{V}_m^{\left(i\right)}\right),$ for $m\in {\mathbb{N}}_0$, and $i=1,2,3.$ Using these iterations, the values $v_0^{\left(i\right)}$, $v_1^{\left(i\right)}$, $v_2^{\left(i\right)}$, $v_3^{\left(i\right)},$ for $i=1,2,3$ can be explicitly expressed in terms of the powers of the initial variables and the corresponding $V_m^{\left(i\right)}$ for $i=1,2,3$, as follows:

$$v_0^{\left(i\right)}={\displaystyle \frac{{\left(v_{-2}^{\left(i\right)}\right)}^{r_i}}{v_{-1}^{\left(i\right)}{V}_0^{\left(i\right)}}}={\displaystyle \frac{{\left(v_{-2}^{\left(i\right)}\right)}^{r_i{F}_0^{\left(i\right)}}}{{\left(v_{-1}^{\left(i\right)}\right)}^{F_1^{\left(i\right)}}{\left(V_0^{\left(i\right)}\right)}^{F_0^{\left(i\right)}}}},\hspace{1em}\mathrm{for} i=1,2,3.$$

Moreover,

$$v_1^{\left(i\right)}={\displaystyle \frac{{\left(v_{-1}^{\left(i\right)}\right)}^{r_i}}{v_0^{\left(i\right)}{V}_1^{\left(i\right)}}}={\displaystyle \frac{{\left(v_{-1}^{\left(i\right)}\right)}^{r_i+1}{V}_0^{\left(i\right)}}{V_1^{\left(i\right)}{\left(v_{-2}^{\left(i\right)}\right)}^{r_i}}}={\displaystyle \frac{{\left(v_{-1}^{\left(i\right)}\right)}^{F_2^{\left(i\right)}}{\left(V_0^{\left(i\right)}\right)}^{F_1^{\left(i\right)}}}{{\left(V_1^{\left(i\right)}\right)}^{F_0^{\left(i\right)}}{\left(v_{-2}^{\left(i\right)}\right)}^{r_i{F}_1^{\left(i\right)}}}},\hspace{1em}\mathrm{for} i=1,2,3,$$

and

$$v_2^{\left(i\right)}={\displaystyle \frac{{\left(v_0^{\left(i\right)}\right)}^{r_i}}{v_1^{\left(i\right)}{V}_2^{\left(i\right)}}}={\displaystyle \frac{{\left(v_{-2}^{\left(i\right)}\right)}^{r_i+r_i^2}{V}_1^{\left(i\right)}}{{\left(v_{-1}^{\left(i\right)}\right)}^{1+2r_i}{\left(V_0^{\left(i\right)}\right)}^{1+r_i}{V}_2^{\left(i\right)}}}={\displaystyle \frac{{\left(v_{-2}^{\left(i\right)}\right)}^{r_i{F}_2^{\left(i\right)}}{\left(V_1^{\left(i\right)}\right)}^{F_1^{\left(i\right)}}}{{\left(v_{-1}^{\left(i\right)}\right)}^{F_3^{\left(i\right)}}{\left(V_0^{\left(i\right)}\right)}^{F_2^{\left(i\right)}}{\left(V_2^{\left(i\right)}\right)}^{F_0^{\left(i\right)}}}},\hspace{1em}\mathrm{for} i=1,2,3$$

and

$$v_3^{\left(i\right)}={\displaystyle \frac{{\left(v_1^{\left(i\right)}\right)}^{r_i}}{v_2^{\left(i\right)}{V}_3^{\left(i\right)}}}={\displaystyle \frac{{\left(v_{-1}^{\left(i\right)}\right)}^{F_4^{\left(i\right)}}{\left(V_0^{\left(i\right)}\right)}^{F_3^{\left(i\right)}}{\left(V_2^{\left(i\right)}\right)}^{F_1^{\left(i\right)}}}{{\left(V_1^{\left(i\right)}\right)}^{F_2^{\left(i\right)}}{\left(V_3^{\left(i\right)}\right)}^{F_0^{\left(i\right)}}{\left(v_{-2}^{\left(i\right)}\right)}^{r_i{F}_3^{\left(i\right)}}}},\hspace{1em}\mathrm{for} i=1,2,3.$$

By applying the principle of mathematical induction, we derive the following general expressions:

$$\begin{array}{cc}{v}_{2m}^{\left(i\right)}=\hfill & {\displaystyle \frac{{\left(v_{-2}^{\left(i\right)}\right)}^{r_i{F}_{2m}^{\left(i\right)}}}{{\left(v_{-1}^{\left(i\right)}\right)}^{F_{2m+1}^{\left(i\right)}}{\left(V_{2m}^{\left(i\right)}\right)}^{F_0^{\left(i\right)}}}}\left\{{\displaystyle \prod _{k=0}^{m-1}}{\displaystyle \frac{{\left(V_{2k+1}^{\left(i\right)}\right)}^{F_{2(m-k)-1}^{\left(i\right)}}}{{\left(V_{2k}^{\left(i\right)}\right)}^{F_{2(m-k)}^{\left(i\right)}}}}\right\},\hfill \\ v_{2m+1}^{\left(i\right)}=\hfill & {\displaystyle \frac{{\left(v_{-1}^{\left(i\right)}\right)}^{F_{2(m+1)}^{\left(i\right)}}}{{\left(v_{-2}^{\left(i\right)}\right)}^{r_i{F}_{2m+1}^{\left(i\right)}}}}\left\{{\displaystyle \prod _{k=0}^m}{\displaystyle \frac{{\left(V_{2k}^{\left(i\right)}\right)}^{F_{2(m-k)+1}^{\left(i\right)}}}{{\left(V_{2k+1}^{\left(i\right)}\right)}^{F_{2(m-k)}^{\left(i\right)}}}}\right\}, \mathrm{for} m\in {\mathbb{N}}_0, \mathrm{and} i=1,2,3.\hfill \end{array}$$

To complete the proof, we present the induction argument that leads to the closed-form expressions of $v_{2m}^{\left(i\right)}$ and $v_{2m+1}^{\left(i\right)}$. First, the base case $m=0$ is directly verified. Next, assume that the formulas for $v_{2m}^{\left(i\right)}$ and $v_{2m+1}^{\left(i\right)}$ hold for a fixed m. Using the recurrence relation $v_m^{\left(i\right)}={\left(v_{m-2}^{\left(i\right)}\right)}^{r_i}/\left(v_{m-1}^{\left(i\right)}{V}_m^{\left(i\right)}\right)$, substituting the inductive hypothesis, and collecting the multiplicative factors according to their the index structure, we obtain the next two expressions

$$\begin{array}{cc}\hfill v_{2m+2}^{\left(i\right)}& ={\displaystyle \frac{{\left(v_{2m}^{\left(i\right)}\right)}^{r_i}}{v_{2m+1}^{\left(i\right)}{V}_{2m+2}^{\left(i\right)}}}\hfill \\ & ={\displaystyle \frac{{\left({\displaystyle \frac{{\left(v_{-2}^{\left(i\right)}\right)}^{r_i{F}_{2m}^{\left(i\right)}}}{{\left(v_{-1}^{\left(i\right)}\right)}^{F_{2m+1}^{\left(i\right)}}{\left(V_{2m}^{\left(i\right)}\right)}^{F_0^{\left(i\right)}}}}\left\{{\displaystyle \prod _{k=0}^{m-1}}{\displaystyle \frac{{\left(V_{2k+1}^{\left(i\right)}\right)}^{F_{2(m-k)-1}^{\left(i\right)}}}{{\left(V_{2k}^{\left(i\right)}\right)}^{F_{2(m-k)}^{\left(i\right)}}}}\right\}\right)}^{r_i}}{\left({\displaystyle \frac{{\left(v_{-1}^{\left(i\right)}\right)}^{F_{2(m+1)}^{\left(i\right)}}}{{\left(v_{-2}^{\left(i\right)}\right)}^{r_i{F}_{2m+1}^{\left(i\right)}}}}\left\{{\displaystyle \prod _{k=0}^m}{\displaystyle \frac{{\left(V_{2k}^{\left(i\right)}\right)}^{F_{2(m-k)+1}^{\left(i\right)}}}{{\left(V_{2k+1}^{\left(i\right)}\right)}^{F_{2(m-k)}^{\left(i\right)}}}}\right\}\right)V_{2m+2}^{\left(i\right)}}}\hfill \\ & ={\displaystyle \frac{{\left(v_{-2}^{\left(i\right)}\right)}^{r_i{F}_{2m+1}^{\left(i\right)}+r_i^2{F}_{2m}^{\left(i\right)}}}{{\left(v_{-1}^{\left(i\right)}\right)}^{F_{2(m+1)}^{\left(i\right)}+r_i{F}_{2m+1}^{\left(i\right)}}{V}_{2m+2}^{\left(i\right)}}}{\displaystyle \frac{{\left(V_{2m+1}^{\left(i\right)}\right)}^{F_1^{\left(i\right)}}}{{\left(V_{2m}^{\left(i\right)}\right)}^{F_1^{\left(i\right)}+r_i{F}_0^{\left(i\right)}}}}\left\{{\displaystyle \prod _{k=0}^{m-1}}{\displaystyle \frac{{\left(V_{2k+1}^{\left(i\right)}\right)}^{F_{2(m-k)}^{\left(i\right)}+r_i{F}_{2(m-k)-1}^{\left(i\right)}}}{{\left(V_{2k}^{\left(i\right)}\right)}^{F_{2(m-k)+1}^{\left(i\right)}+r_i{F}_{2(m-k)}^{\left(i\right)}}}}\right\}\hfill \\ & ={\displaystyle \frac{{\left(v_{-2}^{\left(i\right)}\right)}^{r_i{F}_{2m+2}^{\left(i\right)}}}{{\left(v_{-1}^{\left(i\right)}\right)}^{F_{2m+3}^{\left(i\right)}}{\left(V_{2m+2}^{\left(i\right)}\right)}^{F_0^{\left(i\right)}}}}\left\{{\displaystyle \prod _{k=0}^m}{\displaystyle \frac{{\left(V_{2k+1}^{\left(i\right)}\right)}^{F_{2(m-k)+1}^{\left(i\right)}}}{{\left(V_{2k}^{\left(i\right)}\right)}^{F_{2(m-k)+2}^{\left(i\right)}}}}\right\},\hfill \end{array}$$

$$\begin{array}{cc}\hfill v_{2m+3}^{\left(i\right)}& ={\displaystyle \frac{{\left(v_{2m+1}^{\left(i\right)}\right)}^{r_i}}{v_{2m+2}^{\left(i\right)}{V}_{2m+3}^{\left(i\right)}}}\hfill \\ & ={\displaystyle \frac{{\left({\displaystyle \frac{{\left(v_{-1}^{\left(i\right)}\right)}^{F_{2(m+1)}^{\left(i\right)}}}{{\left(v_{-2}^{\left(i\right)}\right)}^{r_i{F}_{2m+1}^{\left(i\right)}}}}\left\{{\displaystyle \prod _{k=0}^m}{\displaystyle \frac{{\left(V_{2k}^{\left(i\right)}\right)}^{F_{2(m-k)+1}^{\left(i\right)}}}{{\left(V_{2k+1}^{\left(i\right)}\right)}^{F_{2(m-k)}^{\left(i\right)}}}}\right\}\right)}^{r_i}}{\left({\displaystyle \frac{{\left(v_{-2}^{\left(i\right)}\right)}^{r_i{F}_{2m+2}^{\left(i\right)}}}{{\left(v_{-1}^{\left(i\right)}\right)}^{F_{2m+3}^{\left(i\right)}}{\left(V_{2m+2}^{\left(i\right)}\right)}^{F_0^{\left(i\right)}}}}\left\{{\displaystyle \prod _{k=0}^m}{\displaystyle \frac{{\left(V_{2k+1}^{\left(i\right)}\right)}^{F_{2(m-k)+1}^{\left(i\right)}}}{{\left(V_{2k}^{\left(i\right)}\right)}^{F_{2(m-k)+2}^{\left(i\right)}}}}\right\}\right)V_{2m+3}^{\left(i\right)}}}\hfill \\ & ={\displaystyle \frac{{\left(v_{-1}^{\left(i\right)}\right)}^{F_{2m+3}^{\left(i\right)}+r_i{F}_{2(m+1)}^{\left(i\right)}}}{{\left(v_{-2}^{\left(i\right)}\right)}^{r_i{F}_{2m+2}^{\left(i\right)}+r_i^2{F}_{2m+1}^{\left(i\right)}}}}{\displaystyle \frac{{\left(V_{2m+2}^{\left(i\right)}\right)}^{F_1^{\left(i\right)}}}{{\left(V_{2m+3}^{\left(i\right)}\right)}^{F_0^{\left(i\right)}}}}\left\{{\displaystyle \prod _{k=0}^m}{\displaystyle \frac{{\left(V_{2k}^{\left(i\right)}\right)}^{F_{2(m-k)+2}^{\left(i\right)}+r_i{F}_{2(m-k)+1}^{\left(i\right)}}}{{\left(V_{2k+1}^{\left(i\right)}\right)}^{F_{2(m-k)+1}^{\left(i\right)}+r_i{F}_{2(m-k)}^{\left(i\right)}}}}\right\}\hfill \\ & ={\displaystyle \frac{{\left(v_{-1}^{\left(i\right)}\right)}^{F_{2(m+2)}^{\left(i\right)}}}{{\left(v_{-2}^{\left(i\right)}\right)}^{r_i{F}_{2m+3}^{\left(i\right)}}}}\left\{{\displaystyle \prod _{k=0}^{m+1}}{\displaystyle \frac{{\left(V_{2k}^{\left(i\right)}\right)}^{F_{2(m-k)+3}^{\left(i\right)}}}{{\left(V_{2k+1}^{\left(i\right)}\right)}^{F_{2(m-k)+2}^{\left(i\right)}}}}\right\}.\hfill \end{array}$$

To streamline the presentation and avoid repeating six structurally identical expressions, we reorganize the induction formulas into the unified patterns $v_{6m+2\tau }^{\left(i\right)}$ and $v_{6m+2\tau +1}^{\left(i\right)}$, for $\tau =0,1,2$, which simply group together the six recurrence behaviors appearing over one full cycle of length six. Thus, we have established the general solution formula through iterative analysis and linear transformation, highlighting the recursive structure connecting the original and newly introduced variables. This completes the proof. □

When analyzing iterative nonlinear systems, examining the special case of constant coefficients is a fundamental step toward understanding the system’s long-term dynamic behavior. In this context, considering the case where all coefficients of system (2) are constant allows for a simplification of the general closed-form solution formulas and provides insights into the growth behavior of the system’s internal variables. This result is formalized in the following corollary.

Corollary 1.

Assume that all coefficients of the nonlinear system (2) are constant, i.e., $a_m^{\left(i\right)}=a_1^{\left(i\right)}$ and $b_m^{\left(i\right)}=b_1^{\left(i\right)},$ for $m\in {\mathbb{N}}_0$ and $i=1,2,3.$ Assume further that the initial values $v_{-\tau }^{\left(i\right)}\in \mathbb{R}$ for $\tau =0,1,2$ and $i=1,2,3$ satisfy the regularity condition (4). Then, the system admits an explicit closed-form solution as given in (5), where the auxiliary sequences ${\{V_m^{\left(i\right)},i=1,2,3\}}_{m\in {\mathbb{N}}_0}$ are characterized according to the following cases:

i. 

If $\left\{{\displaystyle \prod _{k=0}^2}{a}_1^{\left(\right(k\left)\mathrm{mod}3\right)}\right\}=1,$ then the sequences ${\{V_m^{\left(i\right)},i=1,2,3\}}_{m\in {\mathbb{N}}_0}$ grow linearly with m and are given by:

$$V_{3m+\tau }^{\left(i\right)}={\left(v_{\tau -2}^{\left(i\right)}\right)}^{r_i}/\left(v_{\tau -1}^{\left(i\right)},v_{\tau }^{\left(i\right)}\right).+m{\displaystyle \sum _{l=0}^2}\left\{{\displaystyle \prod _{k=0}^{l-1}}{a}_1^{\left(\right(i+k\left)\mathrm{mod}3\right)}\right\}{b}_1^{\left(\right(i+l\left)\mathrm{mod}3\right)},$$

(7)

for all $m\in {\mathbb{N}}_0$, τ$=0,1,2,$ and $i=1,2,3.$

ii. 

If $\left\{{\displaystyle \prod _{k=0}^2}{a}_1^{\left(\right(k\left)\mathrm{mod}3\right)}\right\}\ne 1,$ then the sequences ${\{V_m^{\left(i\right)},i=1,2,3\}}_{m\in {\mathbb{N}}_0}$ exhibit exponential growth according to the formula:

$$\begin{array}{cc}\hfill V_{3m+\tau }^{\left(i\right)}& ={\left\{{\displaystyle \prod _{k=0}^2}{a}_1^{\left(\right(i+k\left)\mathrm{mod}3\right)}\right\}}^m{\left(v_{\tau -2}^{\left(i\right)}\right)}^{r_i}/\left(v_{\tau -1}^{\left(i\right)}{v}_{\tau }^{\left(i\right)}\right)+\left({\left\{{\displaystyle \prod _{k=0}^2}{a}_1^{\left(\right(i+k\left)\mathrm{mod}3\right)}\right\}}^m-1\right)\hfill \\ \hfill & \times {\left(\left\{{\displaystyle \prod _{k=0}^2}{a}_1^{\left(\right(i+k\left)\mathrm{mod}3\right)}\right\}-1\right)}^{-1}{\displaystyle \sum _{l=0}^2}\left\{{\displaystyle \prod _{k=0}^{l-1}}{a}_1^{\left(\right(i+k\left)\mathrm{mod}3\right)}\right\}{b}_1^{\left(\right(i+l\left)\mathrm{mod}3\right)},\hfill \end{array}$$

(8)

for all $m\in {\mathbb{N}}_0$, τ$=0,1,2,$ and $i=1,2,3.$

Proof. 

Assume that all the coefficients of the nonlinear system (2) are constant. Under this assumption, the linear auxiliary system (6) reduces to the following constant-coefficient difference equation:

$$V_{m+3}^{\left(i\right)}=\left\{{\displaystyle \prod _{k=0}^2}{a}_1^{\left(\right(i+k\left)\mathrm{mod}3\right)}\right\}{V}_m^{\left(i\right)}+{\displaystyle \sum _{l=0}^2}\left\{{\displaystyle \prod _{k=0}^{l-1}}{a}_1^{\left(\right(i+k\left)\mathrm{mod}3\right)}\right\}{b}_1^{\left(\right(i+l\left)\mathrm{mod}3\right)},$$

(9)

for $m\in {\mathbb{N}}_0$, and $i=1,2,3.$ By applying Lemma 1, which provides the explicit solutions for linear difference equations with constant coefficients, we distinguish two cases based on the value of the product of the coefficients:

i.

$\left\{{\displaystyle \prod _{k=0}^2}{a}_1^{\left(\right(k\left)\mathrm{mod}3\right)}\right\}=1$, the sequences ${\left\{V_m^{\left(i\right)},i=1,2,3\right\}}_{m\in {\mathbb{N}}_0}$ are given by (7).

ii.

$\left\{{\displaystyle \prod _{k=0}^2}{a}_1^{\left(\right(k\left)\mathrm{mod}3\right)}\right\}\ne 1$, the sequences ${\left\{V_m^{\left(i\right)},i=1,2,3\right\}}_{m\in {\mathbb{N}}_0}$ are given by (8).

Therefore, by applying the closed-form expressions derived from the general solution of linear difference equations with constant coefficients, the result follows directly in both cases. □

Remark 2.

If we assume that the system coefficients and parameters are identical across all indices, namely: $a_m^{\left(i\right)}=a_m,$ $b_m^{\left(i\right)}=b_m,$ $r_i=r,$ and $v_{-\tau }^{\left(i\right)}=v_{-\tau }$ for all $m\in {\mathbb{N}}_0$, $i=1,2,3,$ and $\tau =0,1,2,$ then the nonlinear system (2) undergoes a full dimensional collapse and reduces to a single scalar difference equation of the form:

$$\begin{array}{c}\hfill v_{m+1}={\displaystyle \frac{v_{m-1}^{r+1}}{a_m{v}_{m-2}^r+b_m{v}_m{v}_{m-1}}}, for all m\in {\mathbb{N}}_0.\end{array}$$

This reduction is a direct consequence of imposing complete symmetry on both the parameters and the initial conditions, which synchronizes the evolution of all components and effectively suppresses the multidimensional interactions inherent in the general system. In this particular case, the intricate dynamics of the original three-dimensional system simplify into a tractable one-dimensional recurrence, capturing a reduced yet representative form of the nonlinear behavior. Moreover, the explicit solutions of this scalar equation can be obtained as a direct specialization of the general expression provided in Theorem 1. This observation not only highlights the structural richness of the system under symmetric assumptions but also opens pathways for further analytical investigation of special cases, exact solutions, and qualitative behavior within a simplified framework.

In the study of discrete nonlinear systems, particular attention is devoted to systems whose dynamics are governed by compositions of one-to-one functions. Such systems often exhibit richer structural properties and permit a more explicit characterization of their solutions under suitable conditions. In this context, we focus on a broader class of systems defined by one-to-one functions and analyze their solvability in terms of explicit solutions. Specifically, we consider system (3) with the aim of identifying the conditions under which it admits an explicit solution and of deriving its closed-form expression. The following corollary not only extends our previous findings but also underscores the flexibility of our approach in handling more intricate recursive structures involving invertible functional dependencies.

Corollary 2.

Consider the nonlinear system (3), with initial values $v_{-\tau }^{\left(i\right)}\in \mathbb{R}$ for $i=1,2,3,$ and $\tau =0,1,2$, subject to the admissibility conditions

C.0. 

The non-vanishing condition:

$$\begin{array}{c}{\displaystyle \prod _{i=1}^3}{f}_i\left(v_m^{\left(i\right)}\right){\left(a_m^{\left(i\right)}\left(f_{(i+1)\mathrm{mod}3}(v_{m-2}^{\left(\right(i+1\left)\mathrm{mod}3\right)}\right)\right)}^{r_{(i+1)\mathrm{mod}3}}\hfill \\ +b_m^{\left(i\right)}{f}_{(i+1)\mathrm{mod}3}\left(v_m^{\left(\right(i+1\left)\mathrm{mod}3\right)}\right)f_{(i+1)\mathrm{mod}3}\left(v_{m-1}^{\left(\right(i+1\left)\mathrm{mod}3\right)})\right)\ne 0, for m\in {\mathbb{N}}_0, and i=1,2,3.\hfill \end{array}$$

C.1. 

The domain admissibility condition:

$$\begin{array}{c}{f}_{(i+1)\mathrm{mod}3}\left(v_m^{\left(\right(i+1\left)\mathrm{mod}3\right)}\right)f_{(i+1)\mathrm{mod}3}\left(v_{m-1}^{\left(\right(i+1\left)\mathrm{mod}3\right)}\right){\left(f_i\left(v_{m-1}^{\left(i\right)}\right)\right)}^{r_i}\hfill \\ \times {\left(f_i\left(v_m^{\left(i\right)}\right)\right)}^{-1}{\left(a_m^{\left(i\right)}(f_{(i+1)\mathrm{mod}3}\left(v_{m-2}^{\left(\right(i+1\left)\mathrm{mod}3\right)}\right)\right)}^{r_{(i+1)\mathrm{mod}3}}\hfill \\ +b_m^{\left(i\right)}{f}_{(i+1)\mathrm{mod}3}\left(v_m^{\left(\right(i+1\left)\mathrm{mod}3\right)}\right)f_{(i+1)\mathrm{mod}3}{\left(v_{m-1}^{\left(\right(i+1\left)\mathrm{mod}3\right)})\right)}^{-1}\in D_{f_i^{-1}}, for m\in {\mathbb{N}}_0, and i=1,2,3.\hfill \end{array}$$

Then, under these conditions, the nonlinear system (3) admits explicit closed-form solutions expressed by formula:

$$\begin{array}{cc}\hfill v_{6m+2\tau }^{\left(i\right)}& =f_i^{-1}({\displaystyle \frac{{\left(f_i\left(v_{-2}^{\left(i\right)}\right)\right)}^{r_i{F}_{6m+2\tau }^{\left(i\right)}}}{{\left(f_i\left(v_{-1}^{\left(i\right)}\right)\right)}^{F_{6m+2\tau +1}^{\left(i\right)}}{\left(V_{3(2m+\tau )-\tau }^{\left(i\right)}\right)}^{F_0^{\left(i\right)}}}}\left\{{\displaystyle \prod _{k=0}^{m-1+\left[{\displaystyle \frac{\tau +1}{2}}\right]}}{\displaystyle \frac{{\left(V_{3\left(2k\right)+1}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau )-1}^{\left(i\right)}}}{{\left(V_{3\left(2k\right)}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau )}^{\left(i\right)}}}}\right\}\\ & \times \left\{{\displaystyle \prod _{k=0}^{m-1+\left[{\displaystyle \frac{\tau }{2}}\right]}}{\displaystyle \frac{{\left(V_{3(2k+1)}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau -1)-1}^{\left(i\right)}}}{{\left(V_{3\left(2k\right)+2}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau -1)}^{\left(i\right)}}}}\right\}\left\{{\displaystyle \prod _{k=0}^{m-1}}{\displaystyle \frac{{\left(V_{3(2k+1)+2}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau -2)-1}^{\left(i\right)}}}{{\left(V_{3(2k+1)+1}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau -2)}^{\left(i\right)}}}}\right\}),\\ v_{6m+2\tau +1}^{\left(i\right)}\hfill & =f_i^{-1}({\displaystyle \frac{{\left(f_i\left(v_{-1}^{\left(i\right)}\right)\right)}^{F_{2(3m+\tau +1)}^{\left(i\right)}}}{{\left(f_i\left(v_{-2}^{\left(i\right)}\right)\right)}^{r_i{F}_{2(3m+\tau )+1}^{\left(i\right)}}}}\left\{{\displaystyle \prod _{k=0}^m}{\displaystyle \frac{{\left(V_{3\left(2k\right)}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau )+1}^{\left(i\right)}}}{{\left(V_{3\left(2k\right)+1}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau )}^{\left(i\right)}}}}\right\}\hfill \\ & \times \left\{{\displaystyle \prod _{k=0}^{m-1+\left[{\displaystyle \frac{\tau +1}{2}}\right]}}{\displaystyle \frac{{\left(V_{3\left(2k\right)+2}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau -1)+1}^{\left(i\right)}}}{{\left(V_{3(2k+1)}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau -1)}^{\left(i\right)}}}}\right\}\left\{{\displaystyle \prod _{k=0}^{m-1+\left[{\displaystyle \frac{\tau }{2}}\right]}}{\displaystyle \frac{{\left(V_{3(2k+1)+1}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau -2)+1}^{\left(i\right)}}}{{\left(V_{3(2k+1)+2}^{\left(i\right)}\right)}^{F_{2\left(3\right(m-k)+\tau -2)}^{\left(i\right)}}}}\right\}),\hfill \end{array}$$

(10)

for all $m\in {\mathbb{N}}_0$, τ$=0,1,2,$ and $i=1,2,3$, where [x] denotes the integer part (floor) of x, the sequences ${\{V_m^{\left(i\right)},i=1,2,3\}}_{m\in {\mathbb{N}}_0}$, are given by:

$$\begin{array}{cc}\hfill V_{3m+\tau }^{\left(i\right)}=& \left\{{\displaystyle \prod _{k^{\prime }=0}^{m-1}}\left\{{\displaystyle \prod _{k=0}^2}{a}_{3k^{\prime }+\tau +2-k}^{\left(\right(i+k\left)\mathrm{mod}3\right)}\right\}\right\}{\left(f_i\left(v_{\tau -2}^{\left(i\right)}\right)\right)}^{r_i}/\left(f_i\left(v_{\tau -1}^{\left(i\right)}\right)f_i\left(v_{\tau }^{\left(i\right)}\right)\right)\hfill \\ & +{\displaystyle \sum _{l^{\prime }=0}^{m-1}}\left\{{\displaystyle \prod _{k^{\prime }=l^{\prime }+1}^{m-1}}\left\{{\displaystyle \prod _{k=0}^2}{a}_{3k^{\prime }+\tau +2-k}^{\left(\right(i+k\left)\mathrm{mod}3\right)}\right\}\right\}{\displaystyle \sum _{l=0}^2}\left\{{\displaystyle \prod _{k=0}^{l-1}}{a}_{3l^{\prime }+\tau +2-k}^{\left(\right(i+k\left)\mathrm{mod}3\right)}\right\}{b}_{3l^{\prime }+\tau +2-l}^{\left(\right(i+l\left)\mathrm{mod}3\right)},\hfill \end{array}$$

for all $m\in {\mathbb{N}}_0$, τ$=0,1,2,$ and $i=1,2,3.$ Furthermore when all the coefficients are constant, then the nonlinear system (3) admits an explicit closed-form solution (10), where the structure of ${\{V_m^{\left(i\right)},i=1,2,3\}}_{m\in {\mathbb{N}}_0}$ simplifies remarkably, allowing for two distinct cases:

i. 

If $\left\{{\displaystyle \prod _{k=0}^2}{a}_1^{\left(\right(k\left)\mathrm{mod}3\right)}\right\}=1,$ then the sequences ${\left\{V_m^{\left(i\right)},i=1,2,3\right\}}_{m\in {\mathbb{N}}_0}$ are given by:

$$V_{3m+\tau }^{\left(i\right)}={\left(f_i\left(v_{\tau -2}^{\left(i\right)}\right)\right)}^{r_i}/\left(f_i\left(v_{\tau -1}^{\left(i\right)}\right)f_i\left(v_{\tau }^{\left(i\right)}\right)\right)+m{\displaystyle \sum _{l=0}^2}\left\{{\displaystyle \prod _{k=0}^{l-1}}{a}_1^{\left(\right(i+k\left)\mathrm{mod}3\right)}\right\}{b}_1^{\left(\right(i+l\left)\mathrm{mod}3\right)},$$

for all $m\in {\mathbb{N}}_0$, τ$=0,1,2,$ and $i=1,2,3.$

ii. 

If $\left\{{\displaystyle \prod _{k=0}^2}{a}_1^{\left(\right(k\left)\mathrm{mod}3\right)}\right\}\ne 1$, then the sequences ${\{V_m^{\left(i\right)},i=1,2,3\}}_{m\in {\mathbb{N}}_0}$ are given by:

$$\begin{array}{cc}\hfill V_{3m+\tau }^{\left(i\right)}& ={\left\{{\displaystyle \prod _{k=0}^2}{a}_1^{\left(\right(i+k\left)\mathrm{mod}3\right)}\right\}}^m{\left(f_i\left(v_{\tau -2}^{\left(i\right)}\right)\right)}^{r_i}/\left(f_i(v_{\tau -1}^{\left(i\right)}\right)f_i\left(v_{\tau }^{\left(i\right)}\right)+\left({\left\{{\displaystyle \prod _{k=0}^2}{a}_1^{\left(\right(i+k\left)\mathrm{mod}3\right)}\right\}}^m-1\right)\hfill \\ \hfill & \times {\left(\left\{{\displaystyle \prod _{k=0}^2}{a}_1^{\left(\right(i+k\left)\mathrm{mod}3\right)}\right\}-1\right)}^{-1}{\displaystyle \sum _{l=0}^2}\left\{{\displaystyle \prod _{k=0}^{l-1}}{a}_1^{\left(\right(i+k\left)\mathrm{mod}3\right)}\right\}{b}_1^{\left(\right(i+l\left)\mathrm{mod}3\right)},\hfill \end{array}$$

for all $m\in {\mathbb{N}}_0$, τ$=0,1,2,$ and $i=1,2,3.$

Proof. 

Since the functions $f_i,$ for $i=1,2,3$, are continuous and one-to-one, we can apply them directly to the system’s recursive relations. This leads to the following functional equalities:

$$\begin{array}{c}{f}_1\left(v_{m+1}^{\left(1\right)}\right)={\displaystyle \frac{f_2\left(v_m^{\left(2\right)}\left)f_2\right(v_{m-1}^{\left(2\right)}\right){\left(f_1\left(v_{m-1}^{\left(1\right)}\right)\right)}^{r_1}}{f_1\left(v_m^{\left(1\right)}\right){\left(a_m^{\left(1\right)}(f_2\left(v_{m-2}^{\left(2\right)}\right)\right)}^{r_2}+b_m^{\left(1\right)}{f}_2\left(v_m^{\left(2\right)}\right)f_2\left(v_{m-1}^{\left(2\right)})\right)}},\hfill \\ f_2\left(v_{m+1}^{\left(2\right)}\right)={\displaystyle \frac{f_3\left(v_m^{\left(3\right)}\left)f_3\right(v_{m-1}^{\left(3\right)}\right){\left(f_2\left(v_{m-1}^{\left(2\right)}\right)\right)}^{r_2}}{f_2\left(v_m^{\left(2\right)}\right){\left(a_m^{\left(2\right)}(f_3\left(v_{m-2}^{\left(3\right)}\right)\right)}^{r_3}+b_m^{\left(2\right)}{f}_3\left(v_m^{\left(3\right)}\right)f_3\left(v_{m-1}^{\left(3\right)})\right)}},\hfill \\ f_3\left(v_{m+1}^{\left(3\right)}\right)={\displaystyle \frac{f_1\left(v_m^{\left(1\right)}\right)f_1\left(v_{m-1}^{\left(1\right)}\right){\left(f_3\left(v_{m-1}^{\left(3\right)}\right)\right)}^{r_3}}{f_3\left(v_m^{\left(3\right)}\right){\left(a_m^{\left(3\right)}(f_1\left(v_{m-2}^{\left(1\right)}\right)\right)}^{r_1}+b_m^{\left(3\right)}{f}_1\left(v_m^{\left(1\right)}\right)f_1\left(v_{m-1}^{\left(1\right)})\right)}}, \mathrm{for} \mathrm{all} m\in {\mathbb{N}}_0.\hfill \end{array}$$

Next, we perform the variable substitution ${\tilde{v}}_m^{\left(i\right)}=f_i\left(v_m^{\left(i\right)}\right),$ for $m\in {\mathbb{N}}_0$ and $i=1,2,3.$ This change of variables transforms the system into the canonical nonlinear form (2), expressed in terms of the new dependent variables ${\tilde{v}}_m^{\left(i\right)},$ for $i=1,2,3.$ By the invertibility of $f_i,$ for $i=1,2,3$, we can express the original variables as $v_m^{\left(i\right)}=f_i^{-1}\left({\tilde{v}}_m^{\left(i\right)}\right),$ for $i=1,2,3$, ensuring the correct interpretation of the solution in the original variable setting. Finally, by invoking Theorem 1, which provides explicit closed-form expressions for the solutions of the transformed system, the proof is completed. □

Remark 3.

In the analysis of the three-dimensional nonlinear system (3), the choice of transformation functions $f_i,$ for $i=1,2,3$ plays a pivotal role in determining both the solvability of the system and the nature of its explicit solutions. The conditions of continuity and one-to-one property imposed on these functions ensure the existence of well-defined inverses and preserve the qualitative behavior of the system under transformation. The impact of the transformation functions’ forms is significant: exponential functions tend to linearize multiplicative relations into additive ones, logarithmic functions perform the reverse operation, and power functions may simplify polynomial-like relations. In practice, selecting suitable transformation functions requires a careful balance between the complexity of the original system, the tractability of the transformed system, and the feasibility of inverting the transformation, all while considering the numerical stability of the resulting solutions. It is worth noting that this methodology is broadly applicable to a wide class of nonlinear multivariate systems. Understanding the interplay between the properties of the transformation functions and the structure of the system provides a powerful analytical tool, not only for achieving exact mathematical solutions but also for enhancing the interpretability and practical relevance of these solutions in applied contexts.

Remark 4.

It is worth noting that System (3) is a structural generalization of System (2). Indeed, by taking the functional transformations as the identity mappings, that is, setting $f_i\left(x\right)=x$, for $i=1,2,3$. System (3) exactly reduces to System (2). This clearly shows that (2) can be regarded as a particular case of (3) within the broader class of structurally transformed systems.

3. Numerical Illustration: Oscillatory Feedback in Capital Accumulation

To demonstrate the versatility of the proposed framework, we consider a discrete-time nonlinear economic model incorporating oscillatory feedback through time-dependent coefficients:

$$K_{t+1} = a_t{K}_t + b_t\sin \left(K_t\right),$$

where $K_t$ is the state variable (e.g., capital, population, or another evolving quantity) at discrete time t. $a_t$ and $b_t$ represent variable productivity and feedback amplitudes, respectively. The sinusoidal nonlinearity introduces bounded periodic effects that emulate cyclical investment behavior and recurrent market fluctuations, while the variability in $a_t$ and $b_t$ reflects dynamic macroeconomic conditions or adaptive policy responses.

Two representative parameter regimes illustrate distinct dynamical behaviors. In the first case, where $a_t=1.1+0.05\sin \left(0.2t\right)$ and $b_t=0.3+0.02\cos \left(0.1t\right)$, the average gain exceeds unity, producing oscillatory yet unbounded capital growth—an analog of sustained expansion punctuated by cyclical deviations. In contrast, when $a_t=0.92+0.06\sin \left(0.15t\right)$ and $b_t=0.25+0.03\cos \left(0.2t\right)$, the system exhibits bounded oscillations around a stable equilibrium level, typifying recurrent boom-and-recession cycles in sub-unitary productivity regimes.

The two parameter regimes exhibit qualitatively distinct behaviors due to the relative magnitude of the average productivity coefficient $a_t$. When the mean of $a_t$ exceeds unity, the linear multiplicative effect dominates the bounded sinusoidal feedback, resulting in oscillatory yet unbounded growth. Conversely, when the mean of $a_t$ lies below one, the damping effect counterbalances the periodic forcing, leading to sustained bounded oscillations around a finite equilibrium level. This distinction is essential, as it demonstrates that the proposed transformation framework accurately captures both amplifying and self-regulating behaviors within the same analytical structure. By encompassing these contrasting dynamical regimes, the method proves robust in modeling nonlinear systems that range from persistent expansion to cyclic equilibrium, thereby validating its generality and applicability to a broad class of discrete dynamical phenomena.

Applying the functional transformation, the nonlinear recursion is mapped into a generalized Fibonacci-type linear system of the form

$$y_{t+1}=p_t{y}_t+q_t{y}_{t-1},$$

where the sequences $\left\{p_t\right\}$ and $\left\{q_t\right\}$ are determined explicitly by the variable coefficients $a_t$ and $b_t$. The resulting closed-form representation

$$y_t=c_1{F}_t+c_2{G}_t$$

yields, through the inverse transformation, an explicit analytical expression for $K_t$. This construction elucidates the intrinsic link between nonlinear oscillatory capital dynamics and generalized Fibonacci recursions, demonstrating that time-dependent trigonometric feedback can be captured precisely within the proposed transformation framework. The example underscores the method’s capacity to describe cyclic, bounded, and expanding behaviors in discrete economic systems while preserving full analytical tractability.

4. Conclusions

This paper presents a comprehensive analysis of a three-dimensional nonlinear difference system characterized by cyclic interdependence among its components. Using systematic transformation techniques, we establish the existence of explicit closed-form solutions for both the original system (2) and its generalized functional form (3). The main contributions of this research lie in the development of a rigorous mathematical framework for solving complex nonlinear difference system variational transformation methods, and in deriving exact solution formulas expressed in terms of generalized Fibonacci sequences, unveiling the intrinsic connection between nonlinear difference systems and linear recurrence relations. Furthermore, these results are extended to a broader class of systems involving associated functional transformations, significantly enhancing the applicability of the proposed solution methodology. The findings open promising avenues for future research, including numerical applications, modeling of environmental interactions in biological systems, and potential extensions to higher-dimensional systems. The methodology developed herein provides a solid foundation for addressing more intricate nonlinear systems arising in both theoretical and applied mathematics. The comparison between growth-dominated and feedback-regulated regimes underscores the versatility of the proposed transformation approach, confirming its effectiveness in capturing a wide spectrum of nonlinear dynamics ranging from persistent expansion to stable cyclic equilibria.