  Generalized synchronization is plausibly the most complex form of
synchronization. Previous studies have revealed the existence of weak or strong
forms of generalized synchronization depending on the multi- or mono-valued
nature of the mapping between the attractors of two unidirectionally-coupled
systems. Generalized synchronization is here obtained by coupling two systems
with a flat control law. Here, we demonstrate that the corresponding
first-return maps can be topologically conjugate in some cases. Conversely, the
response map can foliated while the drive map is not. We describe the
corresponding types of generalized synchronization, explicitly focusing on the
influence of the coupling strength when significantly different dimensions or
dissipation properties characterize the coupled systems. A taxonomy of
generalized synchronization based on these properties is proposed.
