Sudden changes in the dynamical properties of a supercritical fluid model have been found as a function of pressure and temperature (T/T(c) = 2-5 and P/P(c) = 10-10(3)), striving with the notion of a single phase beyond the critical point established by thermodynamics. The sound propagation in the Terahertz frequency region reveals a sharp dynamic crossover between the gas like and the liquid like regimes along several isotherms, which involves, at sufficiently low densities, the interplay between purely acoustic waves and heat waves. Such a crossover allows one to determine a dynamic line in the phase diagram which exhibits a very tight correlation with a number of thermodynamic observables, showing that the supercritical state is remarkably more complex than thought so far.