New algebraically solvable systems of two autonomous firstorder ordinary differential equations with purely quadratic righthand sides
Abstract
We identify many new solvable subcases of the general dynamical system characterized by two autonomous firstorder ordinary differential equations with purely quadratic righthand 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 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 a 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 timedependent coefficients. We also mention that, using a timedependent 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.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 October 2020
 DOI:
 10.1063/5.0011257
 arXiv:
 arXiv:2009.11200
 Bibcode:
 2020JMP....61j2704C
 Keywords:

 Mathematical Physics
 EPrint:
 12 pages, 11 figure