Solvable systems of two coupled first-order ODEs with homogeneous cubic polynomial right-hand sides

TL;DR

This paper derives explicit solutions and conditions for solvability of a class of coupled first-order ODEs with cubic polynomial right-hand sides, including a subclass of isochronous systems with periodic solutions.

Contribution

It provides explicit parameter relations, solvability constraints, and identifies a class of isochronous systems with completely periodic solutions.

Findings

Explicit solution formulas in terms of parameters

Solvability conditions via algebraic constraints

Identification of isochronous cubic systems

Abstract

The solution $x_n\left(t\right)$, $n=1,2,$ of the \textit{initial-values} problem is reported of the \textit{autonomous} system of $2$ coupled first-order ODEs with \textit{homogeneous cubic polynomial} right-hand sides, \begin{eqnarray} \dot{x}_n = c_{n1} \left(x_1\right)^3 + c_{n2}\left( x_1\right)^2 x_2 + c_{n3} x_1 \left(x_2\right)^2+c_{n4} \left(x_2\right)^3\ ,\quad n=1,2\ , \nonumber \end{eqnarray} when the $8$ (time-independent) coefficients $c_{n\ell}$ are appropriately defined in terms of $7$ \textit{arbitrary} parameters, which then also identify the solution of this model. The inversion of these relations is also investigated, namely how to obtain, in terms of the $8$ coefficients $c_{n\ell},$ the $7$ parameters characterizing the solution of this model; and $2$ \textit{constraints} are \textit{explicitly} identified which, if satisfied by the $8$ parameters $c_{n\ell },$ guarantee the \textit{solvability by algebraic operations} of this dynamical system. Also identified is a related, \textit{appropriately modified}, class of (generally \textit{complex}) systems, reading \begin{eqnarray} \dot{{\tilde{x}}_{n}} = \mathbf{i}\omega \tilde{x}_{n} + c_{n1}\left(\tilde{x}_{1}\right) ^{3}+c_{n2}\left( \tilde{x}_{1}\right) ^2 \tilde{x}_2 + c_{n3}\tilde{x}_1 \left( \tilde{x}_2\right)^2 + c_{n4}\left(\tilde{x}_2 \right)^3\ ,\quad n=1,2\ , \nonumber \end{eqnarray} with $\mathbf{i}\omega $ an \textit{arbitrary imaginary} parameter, which feature the remarkable property to be \textit{isochronous}, namely their \textit{generic} solutions are -- as functions of \textit{real time} -- \textit{completely periodic} with a period which is, for each of these models, a \textit{fixed} \textit{integer multiple} of the basic period $\tilde{T}=2\pi /\left\vert \omega \right\vert$.