# Two Peculiar Classes of Solvable Systems Featuring 2 Dependent Variables Evolving in Discrete-Time via 2 Nonlinearly-Coupled First-Order Recursion Relations

## TL;DR

This paper explores specific algebraically solvable discrete-time systems with two dependent variables evolving via nonlinear coupled recursions, extending previous models to include higher-degree homogeneous functions.

## Contribution

It introduces a broader class of solvable systems by generalizing from quadratic to arbitrary degree homogeneous functions in two coupled discrete-time recursions.

## Key findings

- Identified algebraically solvable two-variable discrete systems.
- Extended previous quadratic models to arbitrary degree homogeneous functions.
- Clarified properties and structure of the extended models.

## Abstract

In this paper we identify certain peculiar systems of 2 discrete-time evolution equations,x~n = F^(n)(x1; x2) , n = 1, 2 , which are algebraically solvable. Here l is the "discrete-time" independent variable taking integer values (l = 0, 1, 2,...), xn = xn (l) are 2 dependent variables, and x~n = xn (l + 1) are the corresponding 2 updated variables. In a previous paper the 2 functions F^(n)(x1, x2), n = 1, 2 were defined as follows: F^(n)(x1; x2) = P2 (xn, xn+1), n = 1,2 mod[2]; with P2 (x1; x2) a specific second-degree homogeneous polynomials in the 2 (indistinguishable!) dependent variables x1 (`) and x2 (l). In the present paper we further clarify some aspects of that model and we present its extension to the case when F(n)(x1, x2) = Q^(n)_k(x1, x2), n = 1, 2 mod[2], with Q(n)_k(x1; x2) a specific homogeneous function of arbitrary (integer ) degree k (hence a polynomial of degree k when k > 0) in the 2 dependent variables x1 (l) and x2 (l).