Semi-stable fixed points in plane

Question

Suppose that we have a system of 2 difference equations depending on some parameter and that $(x^*,y^*)$ is non-hyperbolic equilibrium point with one eigenvalue equal to 1 (which appears for the certain parameter value). If we find a center manifold and if the fixed point of the obtained one-dimensional map on center-manifold is semi-stable, what does it mean for the equilibrium of the system i.e. $(x^*,y^*)$? Do we say that it is locally asymptotically stable, unstable or semi-stable (but in this last case, what is semi-stability in the plane)? In the Elaydi's "Discrete Chaos" semi-stable fixed point is in fact unstable (that's how it's classified). I also found in some papers the same thing, but somewhere I found that semi-stable fixed point was considered as the stable one. So, what is correct? Thanks in advance.