= Solution
With $y=\dot x$, the feedback system is
$$
\dot x=y,\qquad \dot y=x^2-(q+x)y-z,\qquad
\dot z=y+p-z-z^3.
$$
For $p=0$, $(x,y,z)=(0,0,0)$ is an <equilibrium>. Its <stability matrix> and characteristic polynomial are
$$
J=\begin{pmatrix}0&1&0\\0&-q&-1\\0&1&-1\end{pmatrix},\qquad
\det(rI-J)=r\bigl(r^2+(q+1)r+q+1\bigr).
$$
Thus \b[there is always a zero <eigenvalue>], with <eigenvector> $(1,0,0)$, so it is a <nonhyperbolic equilibrium>. If $q\ne-1$, the other two <eigenvalues> are off the imaginary axis: for $q>-1$ their real parts are negative, while for $q<-1$ their product is negative. At $q=-1$, all three <eigenvalues> are zero, explaining the exclusion in the next part.
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