Consider the scalar Allen-Cahn equation $$ u_t=u_{xx}+u-u^3. $$ The spatially homogeneous equilibria are $u=0,\pm1$.
$u=0$ is unstable, $u=\pm1$ are stable.
Looking at the phase plane one sees that there are heteroclinic Orbits connecting -1 and +1.
1) What type of equilibria are $u=\pm1$?
Linearization in $u=\pm1$ shows that both equilibria are Center since the linearization matrix has eigenvalues $\pm\sqrt{2}i$ if I am not mistaken.
2) How can there be a heteroclinic Orbit between to centers?
I think there is a sign mistake. As noted in comments, the dynamical system is $$ \begin{cases} u_x = v , \\ v_x = u^3 - u, \end{cases} $$ so its linearization around the equilibria $(u, v)=(\pm 1, 0 )$ is $$ \begin{cases} u_x = v, \\ v_x = 2 u, \end{cases}$$ corresponding to the matrix $\begin{bmatrix} 0 & 1 \\ 2 & 0 \end{bmatrix}$, whose eigenvalues are $\pm \sqrt 2$. So the equilibria are not centers.
EDIT Answer to 2. You say that the phase plane (pag. 3) shows two heteroclinic connecting the stationary solutions $-1$, $+1$. This is not in contradiction because the trajectories correspond to solutions $u_{\pm}$ such that $$\lim_{t\to \pm \infty} (u_{\pm}(t), u'_{\pm}(t))=(\pm 1, 0).$$ So the trajectories do not really "connect" the two stationary solutions, because they do so in infinite time.