$$\dot x = -x + x^3$$ $$\dot y = x + y$$ Where $(x,y) \in \mathbb{R^2}$
I found the fixed points to be:
$$(0,0),(0,1),(0,-1),(1,0),(1,1),(1,-1),(-1,0),(-1,1),(-1,-1)$$
The Jacobian Matrix to be:
\begin{bmatrix} -1+3x^2 & 0\\ 1 & 1 \\ \end{bmatrix}
But every point is unstable.
What am i doing wrong? I feel like:
1) I have too many fixed points and
2) My Jacobian matrix is wrong
Thanks.