  Recently, we have proposed a new dynamic line on the phase diagram in the
supercritical region. Crossing this line corresponds to the radical changes of
the fluid properties. Here, we focus on the dynamics of model Lennard-Jones and
Soft-Sphere fluids. We show that the change of the dynamics from the
liquid-like to gas-like can be established on the basis of the velocity
autocorrelation function and mean-square displacement. Using the rigorous
criterion, we show that the crossover of particle dynamics and key liquid
properties occurs at the same line. We further show that positive sound
dispersion disappears in the vicinity of this line in both kinds of systems.
The dynamic line bears no relationship to the existence of the critical point.
We find that the region of existence of liquid-like dynamics narrows with the
increase of the exponent of the repulsive part of inter-particle potential.
