Global properties of Kuramoto bidirectionally coupled oscillators in a ring structure

Synchronization of Kuramoto model has a notorious interest, since it models phenomena like coordination, synchronization, consensus, averaging, etc. for several systems in biology, physics and engineering. Recently, some steps were done in order to characterize synchronizing interconnection topologies, combining control and algebraic graph theories. In this work, we present a complete analysis of the almost global synchronization property for the class of Kuramoto oscillators bidirectionally coupled in a ring structure. We show that the property is present only up to four oscillators. Although we use Jacobian linearization to prove most of the cases, the particular case of four oscillators requires a more complex approach. We show how this result can be used to characterize some non synchronizing interconnection topologies.