Effect of the initial location of planet d on the expansion process of the Kepler-221 system. Two simulations with identical masses of the planet following mass model M1 (Table 2) and different initial positions are shown from top to bottom. The left panels show the evolution of P_c/P_b and P_e/P_c and the right panels show the evolution of P_d/P_c and P_e/P_d. In the panels, colored dots represent the evolution of the period ratios with the color bar indicating time. The black crosses represent the current period ratio of the system. Dashed lines of different colors correspond to different 3BRs between b, c, and e or c, d, and e, with p, q, and r defined in Equation (2). The black line in panel b represents the analytical evolution of the period ratio only considering b, c, and e 3BR expansion with the position of d fixed.