The Librating Companions in HD 37124, HD 12661, HD 82943, 47 Uma and GJ 876: Alignment or Antialignment?

We investigated the apsidal motion for the multi-planet systems. In the simulations, we found that the two planets of HD 37124, HD 12661, 47 Uma and HD 82943 separately undergo apsidal alignment or antialignment. But the companions of GJ 876 and $\upsilon$ And are only in apsidal lock about $0^{\circ }$. Moreover, we obtained the criteria with Laplace-Lagrange secular theory to discern whether a pair of planets for a certain system are in libration or circulation.