  We propose a model describing $N$ spin-1/2 systems coupled through $N$-order
homogeneous interaction terms, in presence of local time-dependent magnetic
fields. This model can be experimentally implemented with current technologies
in trapped ions and superconducting circuits. By introducing a chain of unitary
transformations, we succeed in exactly converting the quantum dynamics of this
system into that of $2^{N-1}$ fictitious spin-1/2 dynamical problems. We bring
to light the possibility of controlling the unitary evolution of the $N$ spins
generating GHZ states under specific time-dependent scenarios. Moreover, we
show that by appropriately engineering the time-dependence of the coupling
parameters, one may choose a specific subspace in which the $N$-spin system
dynamics takes place. This dynamical feature, which we call time-dependent
selective interaction, can generate a cooling effect of all spins in the
system.
