Location of the (six) fixed points for the Kepler-223 resonant chain (4 planets), as a function of the parameter δ/ Γ (see Sect. 2). I plot the eccentricities of the planets (left), the difference of longitudes of periastron between two successive planets, as well as the two-planet resonant angle σ₄ (center), and the Laplace angles (right). Each fixed point is represented by the same color in all the plots. The gray bands (on the left and right columns) show the observed parameters of the planets (taken from Mills et al. 2016). For the eccentricities (left), the gray bands correspond to the 1σ uncertainties. For the Laplace angles, they correspond to the observed libration amplitudes (which are well constrained by TTVs, see Mills et al. 2016). The observed positions of the planets (and especially the Laplace angles), are consistent with a libration around the red fixed point (see also Fig. 2).