Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 78 / 1 / d / Solution

With $y=\dot{x}$, the feedback system isFor $p=0$, $(x,y,z)=(0,0,0)$ is an equilibrium. Its stability matrix and characteristic polynomial areThus 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.