  Systems evolving under the influence of competing two-body and three-body
interactions, are of particular interest in exploring the stability of
equilibrium states of strongly interacting many-body system. We present a
solvable model based on qubit networks, which allows us to investigate the
intricate influence of these couplings on the possible asymptotic equilibrium
states. We study the asymptotic evolution of finite qubit networks under two
and three-qubit interactions. As representatives of three-qubit interactions we
choose controlled unitary interactions (cu-interactions) with one and two
control qubits. It is shown that networks with purely three-qubit interactions
exhibit different asymptotic dynamics depending on whether we deal with
interactions controlled by one or two qubits. However, when we allow
three-qubit interactions next to two-qubit interactions, the asymptotics is
dictated by two-qubit interactions only. Finally, we prove that the
simultaneous presence of two types of three-qubit interactions results in the
asymptotic dynamics characteristic for two-qubit cu-interactions.
