# Homework Help: Stability of system

1. Dec 14, 2011

### 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)?

2. Dec 15, 2011

### HallsofIvy

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

3. Dec 18, 2011

### namu

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

4. Dec 19, 2011

### mbp

It is not necessary to compute eigenvectors. This system is Hamiltonian (conservative). On dividing one equation by the other you get

\frac{dx}{dy} = \frac{y^2}{x^2}

Separating variables and integrating you find the Hamiltonian

H(x,y) = \frac{1}{3} (x^3-y^3)

The level sets $$H = constant$$ define the phase portrait.

5. Dec 21, 2011

### namu

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