Filippov trajectories and clustering in the Kuramoto model with singular couplingsopen access
- Authors
- Park, Jinyeong; Poyato, David; Soler, Juan
- Issue Date
- 2021
- Publisher
- EUROPEAN MATHEMATICAL SOC-EMS
- Keywords
- Kuramoto models; adaptive coupling; singular interactions; Hebbian learning; Filippov-type solutions; clustering; finite-time synchronization; sticking; Cucker-Smale
- Citation
- JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY, v.23, no.10, pp.3193 - 3278
- Indexed
- SCIE
SCOPUS
- Journal Title
- JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
- Volume
- 23
- Number
- 10
- Start Page
- 3193
- End Page
- 3278
- URI
- https://scholarworks.bwise.kr/hanyang/handle/2021.sw.hanyang/144076
- DOI
- 10.4171/JEMS/1081
- ISSN
- 1435-9855
- Abstract
- We study the synchronization of a generalized Kuramoto system in which the coupling weights are determined by the phase differences between oscillators. We employ the fast-learning regime in a Hebbian-like plasticity rule so that the interaction between oscillators is enhanced by the approach of phases. First, we study the well-posedness problem for the singular weighted Kuramoto systems in which the Lipschitz continuity fails to hold. We present the dynamics of the system equipped with singular weights in all the subcritical, critical and supercritical regimes of the singularity. A key fact is that solutions in the most singular cases must be considered in Filippov's sense. We characterize sticking of phases in the subcritical and critical case and we exhibit a continuation criterion for classical solutions after any collision state in the supercritical regime. Second, we prove that strong solutions to these systems of differential inclusions can be recovered as singular limits of regular weights. We also study the emergence of synchronous dynamics for the singular and regular weighted Kuramoto models.
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