I have the following system of nonlinear polynomial differential equations:
$\begin{cases} \dfrac{dx(t)}{dt}=-3y(t)-3m(2p+x(t)^2-8y(t)^2) \\ \dfrac{dy(t)}{dt}=ex(t)+6mx(t)y(t) \\ \dfrac{dz(t)}{dt}=1+kp + \frac{3}{4}e - 6my(t) \end{cases}$
where $m$, $e$, $p$ and $k$ are constant parameters.
I know that the third equation can not be separated and resolved later by quadrature since $z(t)$ is not present in the first two equations.
The biggest problem is certainly the strong non-linearity in the second equations.
Can anyone tell me how they can solve or the method to solve the first two equations.
Thanks in advance to everyone:)
The first two equations fall into the class of projective Riccati equations, the analysis of which goes back to Lie and Vessiot.
Refer Robert L Anderson, "A Nonlinear Superposition Principle Admitted by Coupled Riccati Equations of the Projective Type", Letters in Mathematical Physics 4 (1980) pp. 1-7.