Time evolution of the bending mode, breathing mode, and surface density in the ring of R = 8–8.5 kpc. First panel: total amplitudes of the bending (solid line) and breathing (dashed line) as functions of time. Second panel: Fourier amplitudes of the bending mode for m = 0, 1, 2, 3, and 4. Different line styles correspond to different Fourier components as indicated in the legend. Third panel: same as the second panel but for the breathing mode. Fourth panel: time evolution of the Fourier amplitudes of the surface density. In all panels, vertical lines indicate the timing of the pericentre passages of the dwarf. The bending mode responds sharply to pericentre passages and decays quickly, while the breathing mode increases more slowly and persists longer, leading to a transition from bending-dominated to breathing-dominated state. The m = 1 and m = 2 are the dominant Fourier components for the bending and breathing modes, respectively.