Undamped periodic forced response is not an attracting steady state

In a finite-dimensional undamped linear oscillator, a bounded periodic particular solution can exist away from natural resonances. Adding any homogeneous normal mode gives another solution, and that free motion does not decay. Thus the late-time motion is not determined by the drive alone. At a resonant frequency with nonzero drive projection onto the corresponding mode, the particular solution grows secularly, so no bounded periodic steady response exists.