Seungjae Lee, Lucas Braun, Frieder Bönisch, Malte Schröder, Moritz Thümler, Marc Timme
{"title":"Complexified Synchrony","authors":"Seungjae Lee, Lucas Braun, Frieder Bönisch, Malte Schröder, Moritz Thümler, Marc Timme","doi":"arxiv-2403.02006","DOIUrl":null,"url":null,"abstract":"The Kuramoto model and its generalizations have been broadly employed to\ncharacterize and mechanistically understand various collective dynamical\nphenomena, especially the emergence of synchrony among coupled oscillators.\nDespite almost five decades of research, many questions remain open, in\nparticular for finite-size systems. Here, we generalize recent work [Phys. Rev.\nLett. 130, 187201 (2023)] on the finite-size Kuramoto model with its state\nvariables analytically continued to the complex domain and also complexify its\nsystem parameters. Intriguingly, systems of two units with purely imaginary\ncoupling do not actively synchronize even for arbitrarily large magnitudes of\nthe coupling strengths, $|K| \\rightarrow \\infty$, but exhibit conservative\ndynamics with asynchronous rotations or librations for all $|K|$. For generic\ncomplex coupling, both, traditional phase-locked states and asynchronous states\ngeneralize to complex locked states, fixed points off the real subspace that\nexist even for arbitrarily weak coupling. We analyze a new collective mode of\nrotations exhibiting finite, yet arbitrarily large winding numbers. Numerical\nsimulations for large networks indicate a novel form of discontinuous phase\ntransition. We close by pointing to a range of exciting questions for future\nresearch.","PeriodicalId":501305,"journal":{"name":"arXiv - PHYS - Adaptation and Self-Organizing Systems","volume":"129 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - PHYS - Adaptation and Self-Organizing Systems","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2403.02006","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
The Kuramoto model and its generalizations have been broadly employed to
characterize and mechanistically understand various collective dynamical
phenomena, especially the emergence of synchrony among coupled oscillators.
Despite almost five decades of research, many questions remain open, in
particular for finite-size systems. Here, we generalize recent work [Phys. Rev.
Lett. 130, 187201 (2023)] on the finite-size Kuramoto model with its state
variables analytically continued to the complex domain and also complexify its
system parameters. Intriguingly, systems of two units with purely imaginary
coupling do not actively synchronize even for arbitrarily large magnitudes of
the coupling strengths, $|K| \rightarrow \infty$, but exhibit conservative
dynamics with asynchronous rotations or librations for all $|K|$. For generic
complex coupling, both, traditional phase-locked states and asynchronous states
generalize to complex locked states, fixed points off the real subspace that
exist even for arbitrarily weak coupling. We analyze a new collective mode of
rotations exhibiting finite, yet arbitrarily large winding numbers. Numerical
simulations for large networks indicate a novel form of discontinuous phase
transition. We close by pointing to a range of exciting questions for future
research.