  In this work, we explore some interesting details of the time-dependent
regime of the long-range systems under mean-field approximation in comparison
with the critical dynamics of the short-range systems. First, we discuss some
mechanisms of the initial anomalous behavior of the magnetization via
two-dimensional Monte Carlo simulations to later compare with results from
Mean-field simulations in both: spin 1/2 (Ising) and spin 1 (Blume Capel model)
Ising models. The distinction between critical and tricritical points is also
investigated. For a complete analysis, we performed short-time simulations in
the mean-field regime to determine the critical temperatures optimizing power
laws and the critical exponents of the different points, which were
independently calculated, i.e. without using previous critical exponents
estimates from literature/theory. Our investigations corroborate analytical
results here also developed.
