Time evolution of the pattern speed for the isolated simulation presented in Appendix A. The intrinsic pattern speed, ${\widehat{\Omega }}_{\text{p}}$ (black line), is obtained from the GALAXY code; the pattern speed is calculated using the DSS method, ${\Omega }_{\text{p}}^{\text{DSS}}$ (red line). We analyzed 95 snapshots from the simulation, and the figure also includes the mean value and corresponding standard deviation of the ratio ${\Omega }_{\text{p}}^{\text{DSS}}/{\widehat{\Omega }}_{\text{p}}$, providing a quantitative comparison of the DSS method accuracy relative to the results from the GALAXY code.