Periodic synchronization and chimera in conformist and contrarian oscillators.

We consider a system of phase oscillators that couple with both attractive and repulsive interaction under a pinning force and explore collective behavior of the system. The oscillators can be divided into two subpopulations of "conformist" oscillators with attractive interaction and "contrarian" ones with repulsive interaction. We find that the interplay between the pinning force and the opposite relationship of the conformist and contrarian oscillators induce peculiar dynamic states: periodic synchronization, breathing chimera, and fully pinned state depending on the fraction of the conformists. Using the Watanabe-Strogatz transformation, we reduce the dynamics into a low-dimensional one and find that the above dynamic states are generated from the reduced dynamics.