Variation of e_b as a functionof ϖ_b − ϖ_c in the absence of damping, showing the existence of low- and high-eccentricity fixed points. Curves produced by integrating the secular equations of motion with the Kepler-419 system parameters as initial conditions except for e_b and ϖ_b − ϖ_c, which are indicated by red dots. In panel a, e_c is held constant, while in panel b, e_c is allowed to vary. Phase curves which cross have different values of e_c at those points. The blue cross shows the position of Kepler-419 and its proximity to the fixed point. We note the different y-axis scales.