# Solvable Dynamical Systems in the Plane with Polynomial Interactions

## TL;DR

This paper presents examples of algebraically solvable two-dimensional dynamical systems with polynomial interactions, using a novel technique involving the evolution of polynomial zeros to identify solvable systems.

## Contribution

Introduces a new method to find algebraically solvable dynamical systems based on polynomial zero evolution, expanding the class of solvable systems.

## Key findings

- Examples of solvable systems with low-degree polynomial interactions
- A new technique for identifying algebraically solvable dynamical systems
- Explicit solutions based on polynomial zero evolution

## Abstract

In this paper we report a few examples of algebraically solvable dynamical systems characterized by 2 coupled Ordinary Differential Equations which read as follows: x_n = P(n) (x1, x2) , n = 1, 2 , with P(n) (x1, x2) specific polynomials of relatively low degree in the 2 dependent variables x1 = x1 (t) and x2 = x2 (t) . These findings are obtained via a new twist of a recent technique to identify dynamical systems solvable by algebraic operations, themselves explicitly identified as corresponding to the time evolutions of the zeros of polynomials the coefficients of which evolve according to algebraically solvable (systems of) evolution equations.