This is basically an exam question, I am supposed to classify the equilibrium points of the following dynamical system:
\begin{equation} \dot{x} = y - x^4 + 1, \qquad \dot{y} = y + x^4 - 1 \end{equation}
It is fairly easy to see that $(1,0)$ and $(-1,0)$ are the only ones. I've read that linearizing only gives an indication of what the equilibrium points might be, and since the system is not Hamiltonian, I'm not really sure how to proceed in the classification. The answer sheet transforms the equilibrium points to the origin, in which they end up with the Jacobian, for example:
\begin{equation} \dot{\begin{pmatrix} x - 1 \\ y \end{pmatrix}} = \begin{pmatrix} -4 & 1 \\ 4 & 1 \end{pmatrix} \cdot \begin{pmatrix} x-1 \\ y \end{pmatrix} \end{equation}, with $\begin{pmatrix} O((x-1)^2) \\ O((x-1)^2)\end{pmatrix}$ as its Big O-term. They further conclude with (1,0) being a saddle due to the eigenvaluees of the linearized matrix. How can one be sure about that?