  We investigate two kinds of topological structures (sphere and torus) spanned
by the controlled parameters of a driven two-level system's Hamiltonian, and
consider the connection between the structures and the system's dynamics. We
discuss the Berry curvature obtained through the dynamical response method,
show the certain physical and observable manifolds including the gapped region
probed by integrating the Berry curvature, and demonstrate the system's state
evolution can be tracked and manipulated by extracting the Berry curvature.
