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.

翻译：我们研究了舆论动力学中周期性行为的涌现及其内在几何结构。为此，我们采用带有逆反个体的有界置信模型，并在卷积社会网络中进行分析。这意味着个体通过在与时变社会网络中的邻居进行互动来调整其观点。由于个体具有逆反性，系统无法达成共识。这一关键特性使得周期性趋势得以涌现。我们证明系统要么收敛至非共识均衡态，要么被吸引至周期或准周期轨道。我们界定了吸引子的维数以及周期性趋势的周期。我们展示了每个轨道在其吸引子内稠密且均匀分布的实例。同时，我们也研究了随机变化社会网络的情形。