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

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 and 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 either is single-valued 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 a single-valued 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, using a time-dependent change of (dependent and independent) variables involving the imaginary parameter iω, isochronous variants of each of the algebraically solvable models we identify, can be explicitly exhibited: by this we mean that these variants all feature the remarkable property that their generic solution is periodic with a period that is a fixed integer multiple of the basic period T = 2π/ω.