{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,12]],"date-time":"2026-09-12T08:43:58Z","timestamp":1789202638565,"version":"build-2803163510"},"reference-count":63,"publisher":"AIP Publishing","issue":"6","license":[{"start":{"date-parts":[[2023,6,5]],"date-time":"2023-06-05T00:00:00Z","timestamp":1685923200000},"content-version":"vor","delay-in-days":4,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2023,6,5]],"date-time":"2023-06-05T00:00:00Z","timestamp":1685923200000},"content-version":"tdm","delay-in-days":4,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["KR1189\/18-2"],"award-info":[{"award-number":["KR1189\/18-2"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2023,6,1]]},"abstract":"<jats:p>In a network of coupled oscillators, a symmetry-broken dynamical state characterized by the coexistence of coherent and incoherent parts can spontaneously form. It is known as a chimera state. We study chimera states in a network consisting of six populations of identical Kuramoto\u2013Sakaguchi phase oscillators. The populations are arranged in a ring, and oscillators belonging to one population are uniformly coupled to all oscillators within the same population and to those in the two neighboring populations. This topology supports the existence of different configurations of coherent and incoherent populations along the ring, but all of them are linearly unstable in most of the parameter space. Yet, chimera dynamics is observed from random initial conditions in a wide parameter range, characterized by one incoherent and five synchronized populations. These observable states are connected to the formation of a heteroclinic cycle between symmetric variants of saddle chimeras, which gives rise to a switching dynamics. We analyze the dynamical and spectral properties of the chimeras in the thermodynamic limit using the Ott\u2013Antonsen ansatz and in finite-sized systems employing Watanabe\u2013Strogatz reduction. For a heterogeneous frequency distribution, a small heterogeneity renders a heteroclinic switching dynamics asymptotically attracting. However, for a large heterogeneity, the heteroclinic orbit does not survive; instead, it is replaced by a variety of attracting chimera states.<\/jats:p>","DOI":"10.1063\/5.0147228","type":"journal-article","created":{"date-parts":[[2023,6,5]],"date-time":"2023-06-05T19:38:25Z","timestamp":1685993905000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":7,"title":["Heteroclinic switching between chimeras in a ring of six oscillator populations"],"prefix":"10.1063","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9750-9991","authenticated-orcid":false,"given":"Seungjae","family":"Lee","sequence":"first","affiliation":[{"name":"Physik-Department, Technische Universit\u00e4t M\u00fcnchen , James-Franck-Stra\u00dfe 1, 85748 Garching, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2664-1127","authenticated-orcid":false,"given":"Katharina","family":"Krischer","sequence":"additional","affiliation":[{"name":"Physik-Department, Technische Universit\u00e4t M\u00fcnchen , James-Franck-Stra\u00dfe 1, 85748 Garching, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2023,6,5]]},"reference":[{"key":"2023081121421828000_c1","volume-title":"Sync","year":"2003"},{"key":"2023081121421828000_c2","volume-title":"Synchronization: A Universal Concept in Nonlinear Sciences","year":"2001"},{"key":"2023081121421828000_c3"},{"key":"2023081121421828000_c4","first-page":"380","article-title":"Coexistence of coherence and incoherence in nonlocally coupled phase oscillators","volume":"5","year":"2002","journal-title":"Nonlinear Phenom. Complex Syst."},{"key":"2023081121421828000_c5","doi-asserted-by":"publisher","first-page":"174102","DOI":"10.1103\/PhysRevLett.93.174102","article-title":"Chimera states for coupled oscillators","volume":"93","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"2023081121421828000_c6","doi-asserted-by":"publisher","first-page":"R67","DOI":"10.1088\/0951-7715\/28\/3\/R67","article-title":"Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators","volume":"28","year":"2015","journal-title":"Nonlinearity"},{"key":"2023081121421828000_c7","doi-asserted-by":"publisher","first-page":"R121","DOI":"10.1088\/1361-6544\/aaaa07","article-title":"The mathematics behind chimera states","volume":"31","year":"2018","journal-title":"Nonlinearity"},{"key":"2023081121421828000_c8","doi-asserted-by":"publisher","first-page":"056125","DOI":"10.1103\/PhysRevE.70.056125","article-title":"Synchronization of two interacting populations of oscillators","volume":"70","year":"2004","journal-title":"Phys. Rev. E"},{"key":"2023081121421828000_c9","doi-asserted-by":"publisher","first-page":"084103","DOI":"10.1103\/PhysRevLett.101.084103","article-title":"Solvable model for chimera states of coupled oscillators","volume":"101","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"2023081121421828000_c10","doi-asserted-by":"publisher","first-page":"012218","DOI":"10.1103\/PhysRevE.93.012218","article-title":"Chimera states in networks of phase oscillators: The case of two small populations","volume":"93","year":"2016","journal-title":"Phys. Rev. E"},{"key":"2023081121421828000_c11","doi-asserted-by":"publisher","first-page":"113101","DOI":"10.1063\/5.0065710","article-title":"Attracting Poisson chimeras in two-population networks","volume":"31","year":"2021","journal-title":"Chaos"},{"key":"2023081121421828000_c12","doi-asserted-by":"publisher","first-page":"093109","DOI":"10.1063\/5.0088465","article-title":"Symmetry breaking yields chimeras in two small populations of Kuramoto-type oscillators","volume":"32","year":"2022","journal-title":"Chaos"},{"key":"2023081121421828000_c13","doi-asserted-by":"publisher","first-page":"042211","DOI":"10.1103\/PhysRevE.100.042211","article-title":"Dynamics and stability of chimera states in two coupled populations of oscillators","volume":"100","year":"2019","journal-title":"Phys. Rev. E"},{"key":"2023081121421828000_c14","doi-asserted-by":"publisher","first-page":"016216","DOI":"10.1103\/PhysRevE.82.016216","article-title":"Bistable chimera attractors on a triangular network of oscillator populations","volume":"82","year":"2010","journal-title":"Phys. Rev. E"},{"key":"2023081121421828000_c15","doi-asserted-by":"publisher","first-page":"043122","DOI":"10.1063\/1.3499502","article-title":"Chimeras in a network of three oscillator populations with varying network topology","volume":"20","year":"2010","journal-title":"Chaos"},{"key":"2023081121421828000_c16","doi-asserted-by":"publisher","first-page":"013121","DOI":"10.1063\/5.0127306","article-title":"Chimeras on a ring of oscillator populations","volume":"33","year":"2023","journal-title":"Chaos"},{"key":"2023081121421828000_c17","doi-asserted-by":"publisher","first-page":"032711","DOI":"10.1103\/PhysRevE.88.032711","article-title":"Stable and flexible system for glucose homeostasis","volume":"88","year":"2013","journal-title":"Phys. Rev. E"},{"key":"2023081121421828000_c18","doi-asserted-by":"publisher","first-page":"264103","DOI":"10.1103\/PhysRevLett.101.264103","article-title":"Partially integrable dynamics of hierarchical populations of coupled oscillators","volume":"101","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"2023081121421828000_c19","doi-asserted-by":"publisher","first-page":"872","DOI":"10.1016\/j.physd.2011.01.002","article-title":"Dynamics of heterogeneous oscillator ensembles in terms of collective variables","volume":"240","year":"2011","journal-title":"Physica D"},{"key":"2023081121421828000_c20","doi-asserted-by":"publisher","first-page":"094819","DOI":"10.1063\/1.4958930","article-title":"Chimera states in two populations with heterogeneous phase-lag","volume":"26","year":"2016","journal-title":"Chaos"},{"key":"2023081121421828000_c21","doi-asserted-by":"publisher","first-page":"011009","DOI":"10.1103\/PhysRevX.4.011009","article-title":"Low-dimensional dynamics of populations of pulse-coupled oscillators","volume":"4","year":"2014","journal-title":"Phys. Rev. X"},{"key":"2023081121421828000_c22","doi-asserted-by":"publisher","first-page":"123125","DOI":"10.1063\/1.4938734","article-title":"Chimera states in coupled Kuramoto oscillators with inertia","volume":"25","year":"2015","journal-title":"Chaos"},{"key":"2023081121421828000_c23","doi-asserted-by":"publisher","first-page":"030901","DOI":"10.1103\/PhysRevE.92.030901","article-title":"Intermittent chaotic chimeras for coupled rotators","volume":"92","year":"2015","journal-title":"Phys. Rev. E"},{"key":"2023081121421828000_c24","doi-asserted-by":"publisher","first-page":"1468","DOI":"10.1088\/0951-7715\/29\/5\/1468","article-title":"Chaotic weak chimeras and their persistence in coupled populations of phase oscillators","volume":"29","year":"2016","journal-title":"Nonlinearity"},{"key":"2023081121421828000_c25","doi-asserted-by":"publisher","first-page":"014102","DOI":"10.1103\/PhysRevLett.117.014102","article-title":"Coherence-resonance chimeras in a network of excitable elements","volume":"117","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"2023081121421828000_c26","doi-asserted-by":"publisher","first-page":"022817","DOI":"10.1103\/PhysRevE.91.022817","article-title":"Chimera states in time-varying complex networks","volume":"91","year":"2015","journal-title":"Phys. Rev. E"},{"key":"2023081121421828000_c27","doi-asserted-by":"publisher","first-page":"043104","DOI":"10.1063\/1.4758814","article-title":"Disorder-induced dynamics in a pair of coupled heterogeneous phase oscillator networks","volume":"22","year":"2012","journal-title":"Chaos"},{"key":"2023081121421828000_c28","doi-asserted-by":"publisher","first-page":"40006","DOI":"10.1209\/0295-5075\/91\/40006","article-title":"Robust features of chimera states and the implementation of alternating chimera states","volume":"91","year":"2010","journal-title":"Europhys. Lett."},{"key":"2023081121421828000_c29","doi-asserted-by":"publisher","first-page":"011044","DOI":"10.1103\/PhysRevX.10.011044","article-title":"Critical switching in globally attractive chimeras","volume":"10","year":"2020","journal-title":"Phys. Rev. X"},{"key":"2023081121421828000_c30","doi-asserted-by":"publisher","first-page":"050201","DOI":"10.1103\/PhysRevE.97.050201","article-title":"Heteroclinic switching between chimeras","volume":"97","year":"2018","journal-title":"Phys. Rev. E"},{"key":"2023081121421828000_c31","doi-asserted-by":"publisher","first-page":"2547","DOI":"10.1007\/s00332-019-09552-5","article-title":"Heteroclinic dynamics of localized frequency synchrony: Heteroclinic cycles for small populations","volume":"29","year":"2019","journal-title":"J. Nonlinear Sci."},{"key":"2023081121421828000_c32","doi-asserted-by":"publisher","first-page":"2571","DOI":"10.1007\/s00332-019-09562-3","article-title":"Heteroclinic dynamics of localized frequency synchrony: Stability of heteroclinic cycles and networks","volume":"29","year":"2019","journal-title":"J. Nonlinear Sci."},{"key":"2023081121421828000_c33","doi-asserted-by":"publisher","first-page":"9883","DOI":"10.1038\/srep09883","article-title":"Self-organized alternating chimera states in oscillatory media","volume":"5","year":"2015","journal-title":"Sci. Rep."},{"key":"2023081121421828000_c34","doi-asserted-by":"publisher","first-page":"071101","DOI":"10.1063\/1.5105367","article-title":"Blinking chimeras in globally coupled rotators","volume":"29","year":"2019","journal-title":"Chaos"},{"key":"2023081121421828000_c35","doi-asserted-by":"publisher","first-page":"2205","DOI":"10.1140\/epjst\/e2020-900270-4","article-title":"Minimal chimera states in phase-lag coupled mechanical oscillators","volume":"229","year":"2020","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"2023081121421828000_c36","doi-asserted-by":"publisher","first-page":"L050204","DOI":"10.1103\/PhysRevE.103.L050204","article-title":"Chimera complexity","volume":"103","year":"2021","journal-title":"Phys. Rev. E"},{"key":"2023081121421828000_c37","doi-asserted-by":"publisher","first-page":"013106","DOI":"10.1063\/1.4905197","article-title":"Weak chimeras in minimal networks of coupled phase oscillators","volume":"25","year":"2015","journal-title":"Chaos"},{"key":"2023081121421828000_c38","doi-asserted-by":"publisher","first-page":"037113","DOI":"10.1063\/1.2930766","article-title":"Low dimensional behavior of large systems of globally coupled oscillators","volume":"18","year":"2008","journal-title":"Chaos"},{"key":"2023081121421828000_c39","doi-asserted-by":"publisher","first-page":"023117","DOI":"10.1063\/1.3136851","article-title":"Long time evolution of phase oscillator systems","volume":"19","year":"2009","journal-title":"Chaos"},{"key":"2023081121421828000_c40","doi-asserted-by":"publisher","first-page":"043104","DOI":"10.1063\/1.3247089","article-title":"Identical phase oscillators with global sinusoidal coupling evolve by M\u00f6bius group action","volume":"19","year":"2009","journal-title":"Chaos"},{"key":"2023081121421828000_c41","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1016\/0167-2789(94)90196-1","article-title":"Constants of motion for superconducting Josephson arrays","volume":"74","year":"1994","journal-title":"Physica D"},{"key":"2023081121421828000_c42","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1186\/s13408-020-00086-9","article-title":"Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: A review","volume":"10","year":"2020","journal-title":"J. Math. Neurosci."},{"key":"2023081121421828000_c43","doi-asserted-by":"publisher","first-page":"1569","DOI":"10.1016\/j.physd.2009.04.012","article-title":"The dynamics of chimera states in heterogeneous Kuramoto networks","volume":"238","year":"2009","journal-title":"Physica D"},{"key":"2023081121421828000_c44","doi-asserted-by":"publisher","first-page":"062221","DOI":"10.1103\/PhysRevE.98.062221","article-title":"Twisted states in low-dimensional hypercubic lattices","volume":"98","year":"2018","journal-title":"Phys. Rev. E"},{"key":"2023081121421828000_c45","doi-asserted-by":"publisher","first-page":"084101","DOI":"10.1103\/PhysRevLett.119.084101","article-title":"Stable chimeras and independently synchronizable clusters","volume":"119","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"2023081121421828000_c46","doi-asserted-by":"publisher","first-page":"4079","DOI":"10.1038\/ncomms5079","article-title":"Cluster synchronization and isolated desynchronization in complex networks with symmetries","volume":"5","year":"2014","journal-title":"Nat. Commun."},{"key":"2023081121421828000_c47","doi-asserted-by":"publisher","first-page":"e1501737","DOI":"10.1126\/sciadv.1501737","article-title":"Complete characterization of the stability of cluster synchronization in complex dynamical networks","volume":"2","year":"2016","journal-title":"Sci. Adv."},{"key":"2023081121421828000_c48","author":"W. R. Inc."},{"key":"2023081121421828000_c49","doi-asserted-by":"publisher","first-page":"044210","DOI":"10.1103\/PhysRevE.106.044210","article-title":"Nontrivial twisted states in nonlocally coupled Stuart-Landau oscillators","volume":"106","year":"2022","journal-title":"Phys. Rev. E"},{"key":"2023081121421828000_c50","volume-title":"Lyapunov Exponents: A Tool to Explore Complex Dynamics","year":"2016"},{"key":"2023081121421828000_c51","first-page":"197","article-title":"A multiplicative ergodic theorem. Characteristic Liapunov, exponents of dynamical systems","volume":"19","year":"1968","journal-title":"Trans. Mosc. Math. Soc."},{"key":"2023081121421828000_c52","doi-asserted-by":"publisher","first-page":"254005","DOI":"10.1088\/1751-8113\/46\/25\/254005","article-title":"Covariant Lyapunov vectors","volume":"46","year":"2013","journal-title":"J. Phys. A: Math. Theor."},{"key":"2023081121421828000_c53","doi-asserted-by":"publisher","first-page":"727","DOI":"10.1007\/s00332-012-9126-5","article-title":"Theory and computation of covariant Lyapunov vectors","volume":"22","year":"2012","journal-title":"J. Nonlinear Sci."},{"key":"2023081121421828000_c54","doi-asserted-by":"publisher","first-page":"054205","DOI":"10.1103\/PhysRevE.107.054205","article-title":"Chaotic chimera attractors in a triangular network of identical oscillators","volume":"107","year":"2023","journal-title":"Phys. Rev. E"},{"key":"2023081121421828000_c55","doi-asserted-by":"publisher","first-page":"103101","DOI":"10.1063\/1.4963371","article-title":"Ott-Antonsen attractiveness for parameter-dependent oscillatory systems","volume":"26","year":"2016","journal-title":"Chaos"},{"key":"2023081121421828000_c56","doi-asserted-by":"publisher","first-page":"013113","DOI":"10.1063\/1.3068353","article-title":"Chimera states in heterogeneous networks","volume":"19","year":"2009","journal-title":"Chaos"},{"key":"2023081121421828000_c57"},{"key":"2023081121421828000_c58","doi-asserted-by":"publisher","first-page":"052215","DOI":"10.1103\/PhysRevE.99.052215","article-title":"Concurrent formation of nearly synchronous clusters in each intertwined cluster set with parameter mismatches","volume":"99","year":"2019","journal-title":"Phys. Rev. E"},{"key":"2023081121421828000_c59"},{"key":"2023081121421828000_c60","doi-asserted-by":"publisher","first-page":"2","DOI":"10.1186\/s13408-015-0033-6","article-title":"Mathematical frameworks for oscillatory network dynamics in neuroscience","volume":"6","year":"2016","journal-title":"J. Math. Neurosci."},{"key":"2023081121421828000_c61","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1016\/j.neuron.2013.12.022","article-title":"The metastable brain","volume":"81","year":"2014","journal-title":"Neuron"},{"key":"2023081121421828000_c62","doi-asserted-by":"publisher","first-page":"60006","DOI":"10.1209\/0295-5075\/86\/60006","article-title":"Sequentially activated groups in neural networks","volume":"86","year":"2009","journal-title":"Europhys. Lett."},{"key":"2023081121421828000_c63","doi-asserted-by":"publisher","first-page":"015003","DOI":"10.1088\/2632-072X\/abbd4c","article-title":"Memory selection and information switching in oscillator networks with higher-order interactions","volume":"2","year":"2021","journal-title":"J. Phys.: Complex."}],"container-title":["Chaos: An Interdisciplinary Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.aip.org\/aip\/cha\/article-pdf\/doi\/10.1063\/5.0147228\/17961397\/063120_1_5.0147228.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/pubs.aip.org\/aip\/cha\/article-pdf\/doi\/10.1063\/5.0147228\/17961397\/063120_1_5.0147228.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.aip.org\/aip\/cha\/article-pdf\/doi\/10.1063\/5.0147228\/17961397\/063120_1_5.0147228.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,7]],"date-time":"2025-05-07T18:54:37Z","timestamp":1746644077000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.aip.org\/cha\/article\/33\/6\/063120\/2894497\/Heteroclinic-switching-between-chimeras-in-a-ring"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,1]]},"references-count":63,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2023,6,1]]}},"URL":"https:\/\/doi.org\/10.1063\/5.0147228","relation":{},"ISSN":["1054-1500","1089-7682"],"issn-type":[{"value":"1054-1500","type":"print"},{"value":"1089-7682","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2023,6]]},"published":{"date-parts":[[2023,6,1]]},"article-number":"063120"}}