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$$\dot x = -x + x^3$$ $$\dot y = x + y$$ 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.

Ananth Kamath
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Jon
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    i have edited the equations in LaTeX . – Ahmad Bazzi Sep 11 '18 at 04:20
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    @Jon: I only find three critical points at $$(x, y) = (-1, 1), (0, 0), (1, -1)$$. The Jacobian is correct and all three critical points are unstable. Draw a phase portrait to see these. – Moo Sep 11 '18 at 04:32
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    @Jon: Stationary points occur when $\dot{x}=0$ and $\dot{y}=0$. You should get $(0,0)$, $(1, -1)$ and $(-1, 1)$. The Jacobian looks correct. – Winter Soldier Sep 11 '18 at 04:33
  • Thanks you both. I see what I did wrong. I appreciate the feedback. – Jon Sep 11 '18 at 04:46

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