  Time dependent mode structure for the Lyapunov vectors associated with the
stepwise structure of the Lyapunov spectra and its relation to the momentum
auto-correlation function are discussed in quasi-one-dimensional many-hard-disk
systems. We demonstrate mode structures (Lyapunov modes) for all components of
the Lyapunov vectors, which include the longitudinal and transverse components
of their spatial and momentum parts, and their phase relations are specified.
These mode structures are suggested from the form of the Lyapunov vectors
corresponding to the zero-Lyapunov exponents. Spatial node structures of these
modes are explained by the reflection properties of the hard-walls used in the
models. Our main interest is the time-oscillating behavior of Lyapunov modes.
It is shown that the largest time-oscillating period of the Lyapunov modes is
twice as long as the time-oscillating period of the longitudinal momentum
auto-correlation function. This relation is satisfied irrespective of the
particle number and boundary conditions. A simple explanation for this relation
is given based on the form of the Lyapunov vector.
