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Hierarchical cluster structures in a one-dimensional swarm oscillator model.

Swarm oscillator model derived by one of the authors (Tanaka), where interacting motile elements form various kinds of patterns, is investigated. We particularly focus on the cluster patterns in one-dimensional space. We mathematically derive all static and stable configurations in final states for a particular but a large set of parameters. In the derivation, we introduce renormalized expression of this model. We find that the static final states are hierarchical cluster structures in which a cluster consists of smaller clusters in a nesting manner.