Time-dependent mode structure for Lyapunov vectors as a collective   movement in quasi-one-dimensional systems
  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. 
Tooru Taniguchi and Gary P. Morriss