  We investigate the dynamics of a two-level system in the presence of an
overall environment composed of two layers. The first layer is just one
single-mode cavity which decays to memoryless reservoir while the second layer
is the two coupled single-mode cavities which decay to memoryless or
memory-keeping reservoirs. In the weak-coupling regime between the qubit and
the first-layer environment, our attention is focused on the effects of the
coupling in the hierarchical environment on the non-Markovian speedup dynamics
behavior of the system. We show that, by controlling the coupling in the
second-layer environment, the multiple dynamics crossovers from Markovian to
non-Markovian and from no-speedup to speedup can be realized. This results hold
independently on the nature of the second-layer environment. Differently, we
find that how the coupling between the two layers affects the non-Markovian
speedup dynamics behavior depends on the nature of the second-layer
environment.
