New algebraically solvable systems of two autonomous first-order ordinary differential equations with purely quadratic right-hand sides

TL;DR

This paper discovers many new algebraically solvable systems of two autonomous quadratic ODEs, expanding the class of solvable dynamical systems with solutions that are elementary or algebraic, including isochronous variants.

Contribution

It identifies numerous new solvable subcases of quadratic autonomous systems, extending classical results and providing explicit algebraic solutions and isochronous variants.

Findings

Many new solvable quadratic systems identified

Solutions are algebraic or singlevalued functions of complex time

Existence of explicit isochronous variants

Abstract

We identify many new solvable subcases of the general dynamical system characterized by two autonomous first-order ordinary differential equations with purely quadratic right-hand sides; the solvable character of these dynamical systems amounting to the possibility to obtain the solution of their initial value problem via algebraic operations. Equivalently---by considering the analytic continuation of these systems to complex time---their algebraically solvable character corresponds to the fact that their general solution is either singlevalued or features only a finite number of algebraic branch points as functions of complex time (the independent variable). Thus our results provide a major enlargement of the class of solvable systems beyond those with singlevalued general solution identified by Garnier about 60 years ago. An interesting property of several of these new dynamical systems is the elementary character of their general solution, identifiable as the roots of a polynomial with explicitly obtainable time-dependent coefficients. We also mention that, via a well-known time-dependent change of (dependent and independent) variables featuring the imaginary parameter $% \mathbf{i} \omega $ (with $\omega $ an arbitrary strictly positive real number), autonomous variants can be explicitly exhibited of each of the algebraically solvable models we identify: variants which all feature the remarkable property to be isochronous, i.e. their generic solution is periodic with a period that is a fixed integer multiple of the basic period $T=2\pi/\omega$.