Here's my system: $$ \left\{ \begin{aligned} \frac{du}{dt} &= u(1-u^2)-w \\[5pt] \frac{dw}{dt} &= u \end{aligned} \right. $$
I'm coming to the conclusion that the only steady state is $(u^*,v^*)=(0,0)$, and that the nullclines are $u=0$ and $w=u(1-u^2)$. Graphing both leads to only one point of intersection, and that is at the origin. Am I wrong?
For linearization, I got the Jacobian to be, $J|_{(0,0)}=\begin{bmatrix} 1&-1 \\ 1&0 \end{bmatrix}$ and that leads to its linearization form, $\begin{bmatrix} \dot{x}\\ \dot{y}\end{bmatrix}=\begin{bmatrix} 1&-1 \\ 1&0 \end{bmatrix}\begin{bmatrix} x\\y\end{bmatrix}$