Three-Variable Differential Equation Stability Discuss the stability of the equilibrium points $(1,0,0)$ and $(1,1,0)$ for the system:
\begin{align}
    x' &= y - y^2\\
    y' &= z\\
    z' &= x - \cos{z}
\end{align}
I have found the jacobian matrix:
$$
        \begin{bmatrix}
        0 & 1 - 2y & 0 \\
        0 & 0 & 1 \\
        1 & 0 & \sin{z} \\
        \end{bmatrix}
$$
and at $(1,0,0)$ this becomes:
$$
        \begin{bmatrix}
        0 & 1 & 0 \\
        0 & 0 & 1 \\
        1 & 0 & 0 \\
        \end{bmatrix}
$$
with characteristic polynomial $1-x^3 = 0$
I am lost at how the eigenvalues can indicate stability from this point; from two-variable systems I can usually make a conclusion at this point but I feel as though I am not taking the right approach to this problem. 
 A: Your last matrix is, basically, a linear transformation on the phase space.
Just like for any other linear transformation, its eigenvectors are the vectors which do not change directions when the transformation is applied to them; moreover, their length is getting multiplied by corresponding eigenvalues. 
Any vector can be represented as a linear combination of eigenvectors of a given "good" linear transformation, thus we can get some information about behavior of an arbitrary vector under this transformation if we know its eigenvalues.
In particular, if your linear transformation (which is, essentially, a linearization of the differentiation operator described by the original system) has real eigenvalues, which are


*

*all positive, then the direction of eigenvectors is preserved, therefore if a vector approaches this point on a phase plane, applying the matrix will not turn it away (so this point is called stable node)

*all negative, then the eigenvectors "change their direction to the opposite", therefore if a vector approaches this point on a phase plane, applying the matrix will turn it around (so this point is called unstable node)

*of a mixed signs, then approaching this point along the eigenvector with positive eigenvalue will result in preserving directions, whereas approaching it form from the direction of eigenvector with negative eigenvalue will result in turning away (this is called saddle point).


There are more complicated cases of complex or repeating eigenvalues, but the basic intuition should be first developed for the described above situations on the phase plane of a system of two ODEs. More information can be found in this Wikipedia article.
