5 - Solvable Dynamical Systems in the Plane with Polynomial Interactions

Summary

In this paper we report a few examples of algebraically solvable dynamical systems characterized by 2 coupled Ordinary Differential Equations which read as follows: $\dot{x}_{n}=P^{\left( n\right) }\left( x_{1},x_{2}\right), n=1,2,$; with $P^{\left( n\right) }\left( x_{1},x_{2}\right)$ specific polynomials of relatively low degree in the 2 dependent variables $x_{1}\equiv x_{1}\left( t\right)$ and $x_{2}\equiv x_{2}\left( t\right)$. 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.