Third-party restoration and annihilation of attractor basins in a dynamical model of coordination*

The Haken-Kelso-Bunz (HKB) equations, as generalized by Zhang to arbitrarily large ensembles of oscillators, model empirical observations in systems exhibiting emergent behavioral and social coordination. The hallmark of the generalized HKB model is its conditional bistability: the existence of stable, rhythmic motions with subpopulations of oscillators clustered in either in-phase or antiphase relative motions for some parameter values, with bifurcations from states exhibiting only in-phase coordination for other values. This work explores a phenomenon whereby a dyad of oscillators, which would be only monostable in isolation, can exhibit bistable coordination when embedded in a larger system. We focus particularly on how breaking symmetry among the oscillators' natural frequencies affects attractor landscapes, yielding quantitative limits on the third-party restoration of dyadic bistability. Our findings illustrate how the whole of a complex system can differ crucially from the sum of its parts. This theoretical work has applications in the social sciences, as well as in healthcare, where it is being used to design interventions aimed at the enhancement of complex behavioral patterns within heterogeneous groups whose members have varying affinities for social coordination.