Stability of Critical Points in a System of DEs

[fluidistic]

Homework Statement

Find the stationary solution(s) of the following system of DE and determine its stability:
x'=x-x^3-xy^2.
y'=2y-y^5-x^4y.

Homework Equations

x'=0, y'=0.
Expansion of x and y around the critical point: x=x_0+\alpha, y=y_0+\beta.

The Attempt at a Solution

I solved x'=y'=0, this gave me x= \pm \sqrt {1-y^2} or x=0. And y=\pm \sqrt {\pm \sqrt {2-x^4}}. Which is likely to give me 12 critical points. Let's consider the (0,0) first...
I expanded x around x_0=y_0=0 via x=x_0+\alpha. I reached that "x'=x-x^3-xy^2" linearizes as \alpha ' \approx \alpha (1-3x_0^2-y_0^2)+x_0-x_0^3-x_0y_0^2-2x_0y_0\beta. Here I see that I'm lucky that I chose first the critical point (0,0) because the linearization reduces to \alpha ' =\alpha and so \alpha =Ae^t. Right here I notice that when t grows up, so does alpha. So that I can already say that (x,y)=(0,0) is not a stable critical point. Is my reasoning right?

P.S.:This is the very first exercise in the assignment of the course dealing with the study of critical points... Looks already quite complicated to me. :/

 

[fzero]

This looks ok so far.

 

[fluidistic]

This looks ok so far.

Ok! Very good to know. Thanks for having taught me how to tackle these kind of problems.