Systems of several first-order quadratic recursions whose evolution is easily ascertainable

TL;DR

This paper derives explicit formulas to solve systems of multiple quadratic recursions with known coefficients, allowing straightforward determination of solutions based on initial parameters.

Contribution

It provides explicit formulas to solve complex quadratic recursion systems, simplifying their analysis and solution process.

Findings

Explicit formulas for solutions of quadratic recursions are derived.

Solutions depend on initial parameters and known coefficients.

The method applies to systems with arbitrary positive integer N.

Abstract

The evolution, as functions of the "ticking time" $\ell =0,1,2,...$, of the solutions of the system of $N$ quadratic recursions \begin{eqnarray*} x_{n}\left( \ell +1\right) =c_{n}+\sum_{m=1}^{N}\left[ C_{nm}x_{m}\left( \ell \right) \right] +\sum_{m=1}^{N}\left\{ d_{nm}\left[ x_{m}\left( \ell \right) \right] ^{2}\right\} +\sum_{m_{1}>m_{2}=1}^{N}\left[ D_{nm_{1}m_{2}}x_{m_{1}}\left( \ell \right) x_{m_{2}}\left( \ell \right) \right] ~,~~~n=1,2,...,N~, && \end{eqnarray*} featuring $N+N^{2}+N^{2}+N\left( N-1\right) N/2=N\left( N+1\right) \left( N+2\right) /2$ ($\ell $-independent) coefficients $c_{n}$, $C_{nm}$, $d_{nm}$ and $D_{nm_{1}m_{2}}$, may be easily ascertained, if these coefficients are given, in terms of $N+N^{2}=N\left( N+1\right) $ a priori arbitrary parameters $a_{n}$ and $b_{nm}$, by $N\left( N+1\right) \left( N+2\right) /2$ explicit formulas provided in this paper. Here $N$ is an arbitrary positive integer.