On certain solvable systems of 4 nonlinear Ordinary Differential Equations

TL;DR

This paper explores specific periodic solutions of a class of four nonlinear ordinary differential equations involving quadratic polynomial interactions among four variables.

Contribution

It presents explicit periodic solutions for a particular solvable system of four nonlinear ODEs with quadratic polynomial terms.

Findings

Derived explicit periodic solutions for the system.

Identified conditions for solvability and periodicity.

Enhanced understanding of nonlinear polynomial ODE systems.

Abstract

Some interesting (periodic!) solutions of certain systems of $4$ nonlinear Ordinary Differential Equations $dx_{n}\left( t\right) /dt=P_{2}^{\left( n\right) }\left[ x_{m}\left( t\right) \right] /\left[ x_{1}\left( t\right) +x_{2}\left( t\right) \right] ,~n,m=1,2,3,4~$, where $P_{2}^{\left( n\right) }\left[ x_{m}\left( t\right) \right] $ are $4$ polynomials of degree $2$ in the $4$ dependent variables $x_{m}\left( t\right) $, are obtained.