# Solvable Systems Featuring 2 Dependent Variables Evolving in Discrete-Time via 2 Nonlinearly-Coupled First-Order Recursion Relations with Polynomial Right-Hand Sides

## TL;DR

This paper analyzes a class of discrete-time systems with two dependent variables evolving through coupled polynomial recursions, extending a continuous-time technique to discrete dynamics with potential applications.

## Contribution

It introduces a modified method to solve coupled polynomial recursion relations in discrete-time, inspired by techniques from continuous-time systems.

## Key findings

- Derived explicit solutions for the coupled polynomial recursions.
- Extended continuous-time solution techniques to discrete-time systems.
- Potential applications in various scientific fields.

## Abstract

The evolution equations mentioned in the title of this paper read as follows: x~n = P(n)(x1; x2) , n = 1, 2 , where l is the "discrete-time" independent variable taking integer values (l =0, 1, 2, ...), xn = xn (l) are the 2 dependent variables, x~n = xn (l + 1), and the 2 functions P(n)(x1, x2), n = 1, 2, are 2 polynomials in the 2 dependent variables x1 (l) and x2 (l). The results reported in this paper have been obtained by an appropriate modification of a recently introduced technique to obtain analogous results in continuous-time t in which case xn = xn (t) and the above recursion relations are replaced by first-order ODEs. Their potential interest is due to the relevance of this kind of evolution equations in various applicative contexts.