  We calculate the dynamic phase transition (DPT) temperatures and present the
dynamic phase diagrams in the Blume-Capel model under the presence of a
time-dependent oscillating external magnetic field by using the path
probability method. We study the time variation of the average order parameters
to obtain the phases in the system and the paramagnetic (P), ferromagnetic (F)
and the F + P mixed phases are found. We also investigate the thermal behavior
of the dynamic order parameters to analyze the nature (continuous and
discontinuous) of transitions and to obtain the DPT points. We present the
dynamic phase diagrams in three planes, namely (T, h), (d, T) and (k2/k1, T),
where T is the reduced temperature, h the reduced magnetic field amplitude, d
the reduced crystal-field interaction and the k2, k1 rate constants. The phase
diagrams exhibit dynamic tricritical and reentrant behaviors as well as a
double critical end point and triple point, strongly depending on the values of
the interaction parameters and the rate constants. We compare and discuss the
dynamic phase diagrams with dynamic phase diagrams that are obtained within the
Glauber-type stochastic dynamics based on the mean-field theory and the
effective field theory.
