I consider the equation $$ u_t=u_{xx}+\cos(u)-1+\Lambda,\text{ with }u\in S^1=\mathbb{R}/2\pi\mathbb{Z}. $$
For $0<\Lambda <2$, there are two spatially homogeneous equilibria (one is stable, the other is unstable) in $S^1$ (lousy speaking, because more precisely they are only two representative elementsts in $S^1$).
For $\Lambda=0$, the equilibria are given by $u=2\pi k, k\in\mathbb{Z}$. So on $S^1$, the only Equilibrium is $u=0$.
Linearizing the System $$ u_x=v,\quad v_x=-\cos(u)+1-\Lambda $$ in $u=0$ gives the linearization matrix $$ \begin{pmatrix}0 & 1\\0&0\end{pmatrix}$$ which has double Zero Eigenvalue.
Same for $\Lambda=2$ and linearization in $u=\pi$.
For $\Lambda <0$ and $\Lambda >2$ we have no equilibria and the Dynamics is oscillatory.
My question(s):
(1) How is this bifurcation called?I did not find the Situation where for the marginal Parameter values (here: $\lambda=0$ and $\Lambda=2$) one has double Zero Eigenvalue.
(2) How to see that for $\Lambda<0$ and $\Lambda>2$ we have oscillatory dynamics?