  We investigate the emergence of periodic behavior in opinion dynamics and its
underlying geometry. For this, we use a bounded-confidence model with
contrarian agents in a convolution social network. This means that agents adapt
their opinions by interacting with their neighbors in a time-varying social
network. Being contrarian, the agents are kept from reaching consensus. This is
the key feature that allows the emergence of cyclical trends. We show that the
systems either converge to nonconsensual equilibrium or are attracted to
periodic or quasi-periodic orbits. We bound the dimension of the attractors and
the period of cyclical trends. We exhibit instances where each orbit is dense
and uniformly distributed within its attractor. We also investigate the case of
randomly changing social networks.
