Symmetry and symmetry breaking in a Kuramoto model induced on a M¨obius strip

摘要

The concept of bounded confidence in social dynamics is introduced into the Kuramoto model. Then a modelnwith projective symmetry is naturally induced by a principal Z2 bundle over a circle. AM¨obius strip is constructednas the manifold of interactions such that two “sides” of a local section correspond to antipodal couplings withnopposite signs.We showanalytically that symmetric polarization of synchronized states (conformist vs contrarian)ncan emerge from the collective behavior of homogeneously coupled oscillators. In the continuum limit, angeneralized asymmetric model reverts to the symmetric case under uniform initial condition, whereas for networksnof finite size, it exhibits richer dynamics or the ordinary behavior of the classic model via symmetry breaking,ndepending on the relevant parameters values. The symmetry and symmetry breaking of the synchronizationnstate are analogous to the concepts of equality and majority of opinions in sociophysical models of opinionnformation. The synchronized clusters can be one (consensus), two (polarization), or more (fragmentation), asnopinion clusters.