System of non-linear ODE's

Question

do you have any suggestions to solve analytically the Non-linear ODE system

$\dot x=18 x^2 y-3p x^2+6p xy$

$\dot y=18 x^2 y-6p xy $

where $p$ is a real constant.

Thank you very much

cheers

Answer 1

Solutions to these types of equations .....(dx/dt) =F1(x,y.z) ;(dy/dt) =F2(x,y,z);(dz/dt) =F3(x,y,z) , ....which are called State Space Equations , are found in Non-linear Control System Engg. and Advanced Control System Engg. books.

Answer 2

There are no explicit solutions for most of the nonlinear systems. Therefore, they are linearized near the fixed points. The eigenvalues near the fixed points mostly give idea about the phase portrait solutions of such systems. For this system fixed points are $(0,0)$ and $(p/3,p/12)$. The eigenvalues near the $(p/3,p/12)$ are:

$\lambda_{1,2}=\frac{-p^{2}}{4}\mp\frac{p^{2}}{2}\sqrt{\frac{31}{4}}i$

which means this is an asymptothycally stable spiral point.

However, the eigenvalues near the origin are found to be zero. This means linearization does not describe the system. Therefore, I don't think you can find an easy solution for this system even when the linearization does not work.