  In this paper, we revisit an interaction problem of two homogeneous
Cucker-Smale (in short C-S) ensembles with attractive-repulsive couplings,
possibly under the effect of Rayleigh friction, and study three sufficient
frameworks leading to bi-cluster flocking in which two sub-ensembles evolve to
two-clusters departing from each other. In previous literature, the interaction
problem has been studied in the context of attractive couplings. In our
interaction problem, inter-ensemble and intra-ensemble couplings are assumed to
be repulsive and attractive respectively. When the Rayleigh frictional forces
are turned on, we show that the total kinetic energy is uniformly bounded so
that spatially mixed initial configurations evolve toward the bi-cluster
configuration asymptotically fast under some suitable conditions on system
parameters, communication weight functions and initial configurations. In
contrast, when Rayleigh frictional forces are turned off, the flocking analysis
is more delicate mainly due to the possibility of an exponential growth of the
kinetic energy. In this case, we employ two mutually disjoint frameworks with
constant inter-ensemble communication function and exponentially localized
inter-ensemble communication functions respectively, and prove the
bi-clustering phenomenon in both cases. This work extends the previous work on
the interaction problem of C-S ensembles. We also conduct several numerical
experiments and compare them with our theoretical results.
