I am a bit unsure about the following problem:
Given:
$\dot{x} = y^2 - 2y + 1$
$\dot{y} = -x^2 + 2x -1$
Find and classify all equilibrium points of the system.
OK, så we know that equilibrium points occur when:
$y^2 - 2y + 1 = 0$
and
$-x^2 + 2x -1 = 0$
It is easy to see that this can only occur at $x = 1, y = 1$.
Now I find the Jacobi matrix for the system:
$J = \begin{bmatrix} 0 & 2y-2 \\ -2x+2 & 0 \end{bmatrix}$
By plotting $x = 1, y = 1$ into the matrix we are left with:
$J = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$
I know that I am now supposed to find the eigenvalues, and from this deduct whether we are dealing with node, spiral, center, etc. But I have never before encoutered a zero matrix in these calcuations before. Basically, if I find the eigenvalue here, I get $\lambda^{2} = 0$, and I don't see how this can tell me anything about the nature of the equilibrium point.
Any help will be truly appreciated!