Finding Eigenvectors & Stabilizing 0,0 in System Stability

[namu]

For the system

\dot{x}=y²
\dot{y}=x²

Both the eigenvalues are zero. How do I
find the eigenvectors so that I can sketch
the phase portrait and how do I classify
the stability of the fixed point (0,0)?

 

[HallsofIvy]

Well, obviously, both x^2 and y^2 are positive for all non-zero x and y so (0, 0) is unstable.

 

[namu]

Yes, that is true. Thank you. How do I find the eigenvectors though?

 

[mbp]

It is not necessary to compute eigenvectors. This system is Hamiltonian (conservative). On dividing one equation by the other you get
\begin{equation}
\frac{dx}{dy} = \frac{y^2}{x^2}
\end{equation}
Separating variables and integrating you find the Hamiltonian
\begin{equation}
H(x,y) = \frac{1}{3} (x^3-y^3)
\end{equation}
The level sets \begin{equation}H = constant\end{equation} define the phase portrait.

 

[namu]

oh my god, that make life so easy. Thank you!