Filippov trajectories and clustering in the Kuramoto model with singular couplings
| Autor: | David Poyato, Juan Soler, Jinyeong Park |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Cucker-Smale
Finite-time synchronization General Mathematics Filippov-type solutions 01 natural sciences Synchronization Adaptive coupling Clustering Continuation Singularity Mathematics - Analysis of PDEs Differential inclusion FOS: Mathematics Applied mathematics 0101 mathematics Mathematics Coupling Applied Mathematics Kuramoto model 010102 general mathematics Kuramoto models Singular interactions State (functional analysis) 70F99 (Primary) 34A12 34A36 34A60 34C15 34D06 92B20 92B25 (Secondary) Lipschitz continuity Sticking Hebbian learning Analysis of PDEs (math.AP) |
| Zdroj: | Digibug: Repositorio Institucional de la Universidad de Granada Universidad de Granada (UGR) Digibug. Repositorio Institucional de la Universidad de Granada instname |
| DOI: | 10.48550/arxiv.1809.04307 |
| Popis: | The work of J. Park was supported by the Basic Research Program through the National Research Foundation of Korea (NRF) funded by the MSIT(NRF-2020R1A4A3079066). The work of D. Poyato (D.P.) and J. Soler (J.S.) has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 639638), the MECD (Spain) research grant FPU14/06304 (D.P), the MINECO-Feder (Spain) research grant number RTI2018-098850-B-I00, the Junta de Andalucia (Spain) Projects PY18-RT-2422 & A-FQM-311-UGR18 (D.P., J.S.). 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. Basic Research Program through the National Research Foundation of Korea (NRF) - MSIT NRF-2020R1A4A3079066 European Research Council (ERC) 639638 MECD (Spain) FPU14/06304 MINECO-Feder (Spain) RTI2018-098850-B-I00 Junta de Andalucia European Commission PY18-RT-2422 A-FQM-311-UGR18 |
| Databáze: | OpenAIRE |
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