  The origin of the bimodality in cluster core entropy is still unknown. At the
same time, recent work has shown that thermal conduction in clusters is likely
a time-variable phenomenon. We consider if time-variable conduction and AGN
outbursts could be responsible for the cool-core (CC), non cool-core (NCC)
dichotomy. We show that strong AGN heating can bring a CC cluster to a NCC
state, which can be stably maintained by conductive heating from the cluster
outskirts. On the other hand, if conduction is shut off by the heat-flux driven
buoyancy instability, then the cluster will cool to the CC state again, where
it is stabilized by low-level AGN heating. Thus, the cluster cycles between CC
and NCC states. In contrast with massive clusters, we predict the CC/NCC
bimodality should vanish in groups, due to the lesser role of conductive
heating there. We find tentative support from the distribution of central
entropy in groups, though firm conclusions require a larger sample carefully
controlled for selection effects.
