Consider $$ u_t=\cos(u)+\mu, u\in S^1=\mathbb{R}/2\pi\mathbb{Z}. $$ for some parameter $\mu$.
For $\mu=1$, the equilibria are $$ u=k\pi. $$ Consider, for example $u=\pi$. Is this stable or unstable?
If I linearize the equation around $u=\pi$, I get $$ u_t=-\sin(\pi)(u-\pi)=0 $$ so the eigenvalues of the linearization are $\lambda_{1,2}=0$.
So this seems to be a non-hyperbolic eigenvalue.
For $\lvert \mu\rvert<1$ there is one stable and one unstable equilibrium and for $\lvert\mu\rvert>1$ there is no equilibrium.
So is the equilibrium for $\mu=1$ a saddle-node equilibrium or how is this called?