{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T17:03:29Z","timestamp":1772211809781,"version":"3.50.1"},"reference-count":50,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2024,11,4]],"date-time":"2024-11-04T00:00:00Z","timestamp":1730678400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,11,4]],"date-time":"2024-11-04T00:00:00Z","timestamp":1730678400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100005765","name":"Universidade de Lisboa","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100005765","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Nonlinear Dyn"],"published-print":{"date-parts":[[2025,3]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>This study examines the synchronization of three identical oscillators arranged in an array and coupled by small impacts, wherein each oscillator interacts solely with its nearest neighbour. The synchronized state, which is asymptotically stable, is characterized by phase opposition among alternating oscillators. We analyse the system using a non-linear discrete dynamical system based on a difference equation derived from the iteration of a plane diffeomorphism. We illustrate these results with the application to a system of three aligned Andronov clocks, showcasing their applicability to a broad range of oscillator systems.<\/jats:p>","DOI":"10.1007\/s11071-024-10524-y","type":"journal-article","created":{"date-parts":[[2024,11,4]],"date-time":"2024-11-04T01:12:32Z","timestamp":1730682752000},"page":"5457-5470","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Huygens synchronization of three aligned clocks"],"prefix":"10.1007","volume":"113","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5444-5089","authenticated-orcid":false,"given":"Jorge","family":"Buescu","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5872-0869","authenticated-orcid":false,"given":"Emma","family":"D\u2019Aniello","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3346-4915","authenticated-orcid":false,"given":"Henrique M.","family":"Oliveira","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,11,4]]},"reference":[{"key":"10524_CR1","first-page":"49","volume-title":"Phase Regulation of Coupled Oscillators and Chaos","author":"R Abraham","year":"1991","unstructured":"Abraham, R.: Phase Regulation of Coupled Oscillators and Chaos, pp. 49\u201378. World Scientific, Singapore (1991)"},{"key":"10524_CR2","unstructured":"Abraham, R., Garfinkel, A.: The dynamics of synchronization and phase regulation. (2003). http:\/\/www.ralph-abraham.org\/articles\/Blurbs\/blurb111.shtml"},{"issue":"1","key":"10524_CR3","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1103\/RevModPhys.77.137","volume":"77","author":"JA Acebr\u00f3n","year":"2005","unstructured":"Acebr\u00f3n, J.A., Bonilla, L.L., P\u00e9rez Vicente, C.J., Ritort, F., Spigler, R.: The kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77(1), 137\u2013185 (2005)","journal-title":"Rev. Mod. Phys."},{"key":"10524_CR4","unstructured":"Andronov, A.A., Vitt, A.A., Khaikin, S.E.: Theory of Oscillators. Pergammon Press, Oxford, New York, 1959\/1963\/1966"},{"issue":"2","key":"10524_CR5","doi-asserted-by":"publisher","first-page":"126","DOI":"10.1016\/0375-9601(94)90947-4","volume":"193","author":"P Ashwin","year":"1994","unstructured":"Ashwin, P., Buescu, J., Stewart, I.: Bubbling of attractors and synchronisation of chaotic oscillators. Phys. Lett. A 193(2), 126\u2013139 (1994)","journal-title":"Phys. Lett. A"},{"issue":"3","key":"10524_CR6","doi-asserted-by":"publisher","first-page":"703","DOI":"10.1088\/0951-7715\/9\/3\/006","volume":"9","author":"P Ashwin","year":"1996","unstructured":"Ashwin, P., Buescu, J., Stewart, I.: From attractor to chaotic saddle: a tale of transverse instability. Nonlinearity 9(3), 703 (1996)","journal-title":"Nonlinearity"},{"key":"10524_CR7","unstructured":"Auslander, M., Buchsbaum, D.: Groups, Rings, Modules. Harper and Row\/Dover, New York, 1974\/2014"},{"key":"10524_CR8","doi-asserted-by":"crossref","unstructured":"Barba-Franco, J., Gallegos, A., Jaimes-Re\u00e1tegui, R., Mu\u00f1oz-Maciel, J., Pisarchik, A.: Dynamics of coexisting rotating waves in unidirectional rings of bistable duffing oscillators. Chaos Interdiscip. J. Nonlinear Sci., 33(7), (2023)","DOI":"10.1063\/5.0141054"},{"issue":"2019","key":"10524_CR9","doi-asserted-by":"publisher","first-page":"563","DOI":"10.1098\/rspa.2001.0888","volume":"458","author":"M Bennett","year":"2002","unstructured":"Bennett, M., Schatz, M., Rockwood, H., Wiesenfeld, K.: Huygen\u2019s clocks. Proc. R. Soc. Lond. Math. Phys. Eng. Sci. 458(2019), 563\u2013579 (2002)","journal-title":"Proc. R. Soc. Lond. Math. Phys. Eng. Sci."},{"issue":"1","key":"10524_CR10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0370-1573(02)00137-0","volume":"366","author":"S Boccaletti","year":"2002","unstructured":"Boccaletti, S., Kurths, J., Osipov, G., Valladares, D., Zhou, C.: The synchronization of chaotic systems. Phys. Rep. 366(1), 1\u2013101 (2002)","journal-title":"Phys. Rep."},{"key":"10524_CR11","doi-asserted-by":"publisher","DOI":"10.1017\/9781107297111","volume-title":"Synchronization: From Coupled Systems to Complex Networks","author":"S Boccaletti","year":"2018","unstructured":"Boccaletti, S., Pisarchik, A.N., del Genio, C.I., Amann, A.: Synchronization: From Coupled Systems to Complex Networks. Cambridge University Press (2018)"},{"issue":"3","key":"10524_CR12","doi-asserted-by":"publisher","first-page":"340","DOI":"10.1038\/nn.4497","volume":"20","author":"M Breakspear","year":"2017","unstructured":"Breakspear, M.: Dynamic models of large-scale brain activity. Nat. Neurosci. 20(3), 340\u2013352 (2017)","journal-title":"Nat. Neurosci."},{"key":"10524_CR13","unstructured":"Buescu, J.: Exotic Attractors: From Liapunov Stability to Riddled Basins, vol. 153. Birkh\u00e4user, (2012)"},{"key":"10524_CR14","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780195301069.001.0001","volume-title":"Rhythms of the Brain","author":"G Buzsaki","year":"2006","unstructured":"Buzsaki, G.: Rhythms of the Brain. Oxford University Press, Oxford (2006)"},{"key":"10524_CR15","unstructured":"Campbell, S.R.: Synchrony and Desynchrony in Neural Oscillators. PhD thesis, The Ohio State University, (1997)"},{"issue":"7","key":"10524_CR16","doi-asserted-by":"publisher","first-page":"1595","DOI":"10.1162\/089976699300016160","volume":"11","author":"SR Campbell","year":"1999","unstructured":"Campbell, S.R., Wang, D.L., Jayaprakash, C.: Synchrony and desynchrony in integrate-and-fire oscillators. Neural Comput. 11(7), 1595\u20131619 (1999)","journal-title":"Neural Comput."},{"issue":"21","key":"10524_CR17","doi-asserted-by":"publisher","first-page":"2047","DOI":"10.1142\/S0218127411029628","volume":"07","author":"K Czolczynski","year":"2011","unstructured":"Czolczynski, K., Perlikowski, P., Stefanski, A., Kapitaniak, T.: Huygen\u2019s odd sympathy experiment revisited. Int. J. Bifurc. Chaos 07(21), 2047\u20132056 (2011)","journal-title":"Int. J. Bifurc. Chaos"},{"issue":"5","key":"10524_CR18","doi-asserted-by":"publisher","first-page":"3303","DOI":"10.1007\/s11071-023-09241-9","volume":"112","author":"E D\u2019Aniello","year":"2024","unstructured":"D\u2019Aniello, E., Oliveira, H.M.: Huygens synchronisation of three clocks equidistant from each other. Nonlinear Dyn. 112(5), 3303\u20133317 (2024)","journal-title":"Nonlinear Dyn."},{"issue":"16","key":"10524_CR19","doi-asserted-by":"publisher","first-page":"L161301","DOI":"10.1103\/PhysRevB.108.L161301","volume":"108","author":"D Dovzhenko","year":"2023","unstructured":"Dovzhenko, D., Aristov, D., Pickup, L., Sigur\u00f0sson, H., Lagoudakis, P.: Next-nearest-neighbor coupling with spinor polariton condensates. Phys. Rev. B 108(16), L161301 (2023)","journal-title":"Phys. Rev. B"},{"issue":"42","key":"10524_CR20","doi-asserted-by":"publisher","first-page":"895","DOI":"10.1016\/j.ijnonlinmec.2007.03.016","volume":"6","author":"A Fradkov","year":"2007","unstructured":"Fradkov, A., Andrievsky, B.: Synchronization and phase relations in the motion of two-pendulum system. Int. J. Non-Linear Mech. 6(42), 895\u2013901 (2007)","journal-title":"Int. J. Non-Linear Mech."},{"key":"10524_CR21","doi-asserted-by":"publisher","DOI":"10.1002\/0471732699","volume-title":"Phaselock Techniques","author":"FM Gardner","year":"2005","unstructured":"Gardner, F.M.: Phaselock Techniques. John Wiley & Sons, New Jersey (2005)"},{"key":"10524_CR22","doi-asserted-by":"crossref","unstructured":"Golubitsky, M., Schaeffer, D., Stewart, I.: Singularities and Groups in Bifurcation. Volume II, volume\u00a069. Applied Mathematical Sciences, (1988)","DOI":"10.1007\/978-1-4612-4574-2"},{"key":"10524_CR23","doi-asserted-by":"crossref","unstructured":"Golubitsky, M., Stewart, I.: Dynamics and Bifurcation in Networks: Theory and Applications of Coupled Differential Equations. Society for Industrial and Applied Mathematics, Philadelphia, PA (2023)","DOI":"10.1137\/1.9781611977332"},{"key":"10524_CR24","unstructured":"Huygens, C.: Letters to de Sluse, Constantyn Huygens, (letters; no. 1333 of 24 February 1665, no. 1335 of 26 February 1665, no. 1345 of 6 March 1665). Societe Hollandaise Des Sciences, Martinus Nijho, La Haye, (1895)"},{"issue":"331","key":"10524_CR25","doi-asserted-by":"publisher","first-page":"2887","DOI":"10.1016\/j.jsv.2012.01.035","volume":"12","author":"V Jovanovic","year":"2012","unstructured":"Jovanovic, V., Koshkin, S.: Synchronization of huygens\u2019 clocks and the poincare method. J. Sound Vib. 12(331), 2887\u20132900 (2012)","journal-title":"J. Sound Vib."},{"issue":"517","key":"10524_CR26","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.physrep.2012.03.002","volume":"1","author":"M Kapitaniak","year":"2012","unstructured":"Kapitaniak, M., Czolczynski, K., Perlikowski, P., Stefanski, A., Kapitaniak, T.: Synchronization of clocks. Phys. Rep. 1(517), 1\u201369 (2012)","journal-title":"Phys. Rep."},{"key":"10524_CR27","doi-asserted-by":"crossref","unstructured":"Kuramoto, Y.: Self-entrainment of a population of coupled non-linear oscillators. In: International Symposium on Mathematical Problems in Theoretical Physics, pp. 420\u2013422. Springer (1975)","DOI":"10.1007\/BFb0013365"},{"key":"10524_CR28","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-69689-3_7","volume-title":"Chemical Turbulence","author":"Y Kuramoto","year":"1984","unstructured":"Kuramoto, Y., Kuramoto, Y.: Chemical Turbulence. Springer, Berlin (1984)"},{"issue":"5","key":"10524_CR29","doi-asserted-by":"publisher","first-page":"1901","DOI":"10.1016\/j.cnsns.2008.07.002","volume":"14","author":"AC Luo","year":"2009","unstructured":"Luo, A.C.: A theory for synchronization of dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 14(5), 1901\u20131951 (2009)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"10524_CR30","doi-asserted-by":"crossref","unstructured":"Luo, A.C.J.: Discrete Systems Synchronization, pp. 197\u2013236. Springer, New York, (2013)","DOI":"10.1007\/978-1-4614-5097-9_6"},{"issue":"110","key":"10524_CR31","doi-asserted-by":"publisher","first-page":"10563","DOI":"10.1073\/pnas.1302880110","volume":"26","author":"EA Martens","year":"2013","unstructured":"Martens, E.A., Thutupalli, S., Fourri\u00e8rec, A., Hallatschek, O.: Chimera states in mechanical oscillator networks. Proc. Natl. Acad. Sci. 26(110), 10563\u201310567 (2013)","journal-title":"Proc. Natl. Acad. Sci."},{"issue":"6","key":"10524_CR32","doi-asserted-by":"publisher","first-page":"1645","DOI":"10.1137\/0150098","volume":"50","author":"RE Mirollo","year":"1990","unstructured":"Mirollo, R.E., Strogatz, S.H.: Synchronization of pulse-coupled biological oscillators. SIAM J. Appl. Math. 50(6), 1645\u20131662 (1990)","journal-title":"SIAM J. Appl. Math."},{"issue":"11548","key":"10524_CR33","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/srep11548","volume":"5","author":"HM Oliveira","year":"2015","unstructured":"Oliveira, H.M., Melo, L.V.: Huygens synchronization of two clocks. Sci. Rep. 5(11548), 1\u201312 (2015). https:\/\/doi.org\/10.1038\/srep11548","journal-title":"Sci. Rep."},{"issue":"10","key":"10524_CR34","doi-asserted-by":"publisher","first-page":"1362","DOI":"10.1080\/10236198.2022.2147001","volume":"28","author":"HM Oliveira","year":"2022","unstructured":"Oliveira, H.M., Perestrelo, S.: Stability of coupled Huygens oscillators. J. Differ. Equ. Appl. 28(10), 1362\u20131380 (2022)","journal-title":"J. Differ. Equ. Appl."},{"key":"10524_CR35","doi-asserted-by":"crossref","unstructured":"Oud, W.T., Nijmeijer, H., Pogromsky, A.Y.: A study of Huijgens\u2019 synchronization: experimental results, volume 336 of Lecture Notes in Control and Information Science, pp. 191\u2013203. Springer, Berlin Heidelberg, (2006)","DOI":"10.1007\/11505532_11"},{"issue":"1","key":"10524_CR36","doi-asserted-by":"publisher","first-page":"1504","DOI":"10.1038\/s41467-017-01190-3","volume":"8","author":"KP O\u2019Keeffe","year":"2017","unstructured":"O\u2019Keeffe, K.P., Hong, H., Strogatz, S.H.: Oscillators that sync and swarm. Nat. Commun. 8(1), 1504 (2017)","journal-title":"Nat. Commun."},{"issue":"8","key":"10524_CR37","doi-asserted-by":"publisher","first-page":"821","DOI":"10.1103\/PhysRevLett.64.821","volume":"64","author":"LM Pecora","year":"1990","unstructured":"Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64(8), 821\u2013824 (1990)","journal-title":"Phys. Rev. Lett."},{"key":"10524_CR38","unstructured":"Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences, volume\u00a012 of Cambridge Nonlinear Science Series. 1st edn., Cambridge University Press, Cambridge, 5 (2003)"},{"key":"10524_CR39","unstructured":"Razavi, B.: Design of Analog CMOS Integrated Circuits. McGraw-Hill, Boston (2005)"},{"issue":"291","key":"10524_CR40","doi-asserted-by":"publisher","first-page":"566","DOI":"10.1016\/j.jsv.2005.06.018","volume":"3\u20135","author":"M Senator","year":"2006","unstructured":"Senator, M.: Synchronization of two coupled escapement-driven pendulum clocks. J. Sound Vib. 3\u20135(291), 566\u2013603 (2006)","journal-title":"J. Sound Vib."},{"key":"10524_CR41","unstructured":"Strogatz, S.: Sync: The Emerging Science of Spontaneous Order. Penguin UK, (2004)"},{"issue":"1\u20134","key":"10524_CR42","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0167-2789(00)00094-4","volume":"143","author":"SH Strogatz","year":"2000","unstructured":"Strogatz, S.H.: From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Phys. D 143(1\u20134), 1\u201320 (2000)","journal-title":"Phys. D"},{"key":"10524_CR43","unstructured":"Strogatz, S.H.: Sync: How order emerges from chaos in the universe, nature, and daily life. Hachette Books, (2012)"},{"issue":"07","key":"10524_CR44","doi-asserted-by":"publisher","first-page":"2450084","DOI":"10.1142\/S0129183124500840","volume":"35","author":"U Uriostegui-Legorreta","year":"2024","unstructured":"Uriostegui-Legorreta, U., Estevez-Delgado, J., P\u00e9rez-Aguilar, H.: Synchronization in network motifs of three piecewise r\u00f6ssler systems coupled in ring configuration. Int. J. Mod. Phys. C 35(07), 2450084 (2024)","journal-title":"Int. J. Mod. Phys. C"},{"issue":"14","key":"10524_CR45","doi-asserted-by":"publisher","first-page":"4303","DOI":"10.1007\/s12648-023-02765-1","volume":"97","author":"U Uriostegui-Legorreta","year":"2023","unstructured":"Uriostegui-Legorreta, U., Tututi, E.: Control and synchronization in the duffing-van der pol and $$\\phi $$ 6 duffing oscillators. Indian J. Phys. 97(14), 4303\u20134315 (2023)","journal-title":"Indian J. Phys."},{"issue":"3","key":"10524_CR46","doi-asserted-by":"publisher","first-page":"1059","DOI":"10.1515\/ijnsns-2022-0465","volume":"24","author":"U Uriostegui-Legorreta","year":"2023","unstructured":"Uriostegui-Legorreta, U., Tututi, E.S.: Master-slave synchronization in the duffing-van der pol and $$\\phi 6$$ duffing oscillators. Int. J. Nonlinear Sci. Numer. Simul. 24(3), 1059\u20131072 (2023)","journal-title":"Int. J. Nonlinear Sci. Numer. Simul."},{"issue":"09","key":"10524_CR47","doi-asserted-by":"publisher","first-page":"2350122","DOI":"10.1142\/S012918312350122X","volume":"34","author":"U Uriostegui-Legorreta","year":"2023","unstructured":"Uriostegui-Legorreta, U., Tututi, E.S.: Numerical study on synchronization in the Rayleigh-Duffing and Duffing oscillators. Int. J. Mod. Phys. C 34(09), 2350122 (2023)","journal-title":"Int. J. Mod. Phys. C"},{"issue":"2","key":"10524_CR48","doi-asserted-by":"publisher","first-page":"356","DOI":"10.1007\/s40435-023-01213-6","volume":"12","author":"U Uriostegui Legorreta","year":"2024","unstructured":"Uriostegui Legorreta, U., Tututi Hern\u00e1ndez, E.S.: Master-slave synchronization in the van der pol-duffing and duffing oscillators. Int. J. Dyn. Control 12(2), 356\u2013372 (2024)","journal-title":"Int. J. Dyn. Control"},{"key":"10524_CR49","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1016\/B978-0-12-068720-6.50022-1","volume-title":"Dynamical Systems and Microphysics: Geometry and Mechanics","author":"J Vassalo-Pereira","year":"1982","unstructured":"Vassalo-Pereira, J.: A theorem on phase-locking in two interacting clocks (the Huygens effect). In: Avez, A., Blaquiere, A., Marzollo, A. (eds.) Dynamical Systems and Microphysics: Geometry and Mechanics, pp. 343\u2013352. Academic Press, New York, London (1982)"},{"key":"10524_CR50","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-22492-2","volume-title":"The Geometry of Biological Time","author":"AT Winfree","year":"1980","unstructured":"Winfree, A.T.: The Geometry of Biological Time. Springer, New York (1980)"}],"container-title":["Nonlinear Dynamics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11071-024-10524-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11071-024-10524-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11071-024-10524-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,2,5]],"date-time":"2025-02-05T22:30:30Z","timestamp":1738794630000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11071-024-10524-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,4]]},"references-count":50,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2025,3]]}},"alternative-id":["10524"],"URL":"https:\/\/doi.org\/10.1007\/s11071-024-10524-y","relation":{"has-preprint":[{"id-type":"doi","id":"10.21203\/rs.3.rs-4401847\/v1","asserted-by":"object"}]},"ISSN":["0924-090X","1573-269X"],"issn-type":[{"value":"0924-090X","type":"print"},{"value":"1573-269X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11,4]]},"assertion":[{"value":"10 May 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 October 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 November 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors report no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}