  We consider a periodically driven system where the high-frequency driving
protocol consists of a sequence of potentials switched on and off at different
instants within a period. We explore the possibility of introducing an
adiabatic modulation of the driving protocol by considering a slow evolution of
the instants when the sequence of potentials is switched on/off. We examine how
this influences the long-term dynamics of periodically driven quantum systems.
By assuming that the slow and fast timescales in the problem can be decoupled,
we derive the stroboscopic (effective) Hamiltonian for a four-step driving
sequence up to the first order in perturbation theory. We then apply this
approach to a rigid rotor, where the adiabatic modulation of the driving
protocol is chosen to produce an evolving emergent magnetic field that
interacts with the rotor's spin. We study the emergence of $\textit{diabolical
points}$ and $\textit{diabolical loci}$ in the parameter space of the effective
Hamiltonian. Further, we study the topological properties of the maps of the
adiabatic paths in the parameter space to the eigenspace of the effective
Hamiltonian. In effect, we obtain a technique to tune the topological
properties of the eigenstates by selecting various adiabatic evolution of the
driving protocol characterized by different paths in the parameter space. This
technique can be applied to any periodic driving protocol to achieve desirable
topological effects.
