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In this paper, we revisit an interaction problem of two homogeneousCucker-Smale (in short C-S) ensembles with attractive-repulsive couplings,possibly under the effect of Rayleigh friction, and study three sufficientframeworks leading to bi-cluster flocking in which two sub-ensembles evolve totwo-clusters departing from each other. In previous literature, the interactionproblem has been studied in the context of attractive couplings. In ourinteraction problem, inter-ensemble and intra-ensemble couplings are assumed tobe repulsive and attractive respectively. When the Rayleigh frictional forcesare turned on, we show that the total kinetic energy is uniformly bounded sothat spatially mixed initial configurations evolve toward the bi-clusterconfiguration asymptotically fast under some suitable conditions on systemparameters, communication weight functions and initial configurations. Incontrast, when Rayleigh frictional forces are turned off, the flocking analysisis more delicate mainly due to the possibility of an exponential growth of thekinetic energy. In this case, we employ two mutually disjoint frameworks withconstant inter-ensemble communication function and exponentially localizedinter-ensemble communication functions respectively, and prove thebi-clustering phenomenon in both cases. This work extends the previous work onthe interaction problem of C-S ensembles. We also conduct several numericalexperiments and compare them with our theoretical results.

Emergent behaviors of the Cucker-Smale ensemble under attractive-repulsive couplings and Rayleigh frictions