{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T23:05:59Z","timestamp":1783983959056,"version":"3.55.0"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T00:00:00Z","timestamp":1778630400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"},{"start":{"date-parts":[[2026,7,14]],"date-time":"2026-07-14T00:00:00Z","timestamp":1783987200000},"content-version":"vor","delay-in-days":62,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12205006"],"award-info":[{"award-number":["12205006"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Excellent Youth Scientific Research Project of Anhui Province","award":["2022AH030107"],"award-info":[{"award-number":["2022AH030107"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Sci Rep"],"DOI":"10.1038\/s41598-026-52344-7","type":"journal-article","created":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T07:34:32Z","timestamp":1778657672000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Turing-driven spiral defect chaos with stationary spiral tips: universal criticality in a time-discrete oscillatory system"],"prefix":"10.1038","volume":"16","author":[{"given":"Qun","family":"Wang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jian","family":"Gao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,5,13]]},"reference":[{"key":"52344_CR1","doi-asserted-by":"publisher","first-page":"634","DOI":"10.1126\/science.175.4022.634","volume":"175","author":"AT Winfree","year":"1972","unstructured":"Winfree, A. T. Spiral waves of chemical activity. Science 175, 634 (1972).","journal-title":"Science"},{"key":"52344_CR2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.123.258102","volume":"123","author":"A Cincotti","year":"2019","unstructured":"Cincotti, A. et al. Threaded rings that swim in excitable media. Phys. Rev. Lett. 123, 258102 (2019).","journal-title":"Phys. Rev. Lett."},{"key":"52344_CR3","doi-asserted-by":"publisher","first-page":"3368","DOI":"10.1103\/PhysRevLett.75.3368","volume":"75","author":"S Grill","year":"1995","unstructured":"Grill, S., Zykov, V. S. & M\u00fcller, S. C. Feedback-controlled dynamics of meandering spiral waves. Phys. Rev. Lett. 75, 3368 (1995).","journal-title":"Phys. Rev. Lett."},{"key":"52344_CR4","doi-asserted-by":"publisher","first-page":"1174","DOI":"10.1103\/PhysRevLett.76.1174","volume":"76","author":"KJ Lee","year":"1996","unstructured":"Lee, K. J., Cox, E. C. & Goldstein, R. E. Competing patterns of signaling activity in dictyostelium discoideum. Phys. Rev. Lett. 76, 1174 (1996).","journal-title":"Phys. Rev. Lett."},{"key":"52344_CR5","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1006\/jtbi.1996.0126","volume":"181","author":"CV Oss","year":"1996","unstructured":"Oss, C. V., Panfilov, A. V., Hogeweg, P., Siegert, F. & Weijer, C. J. Spatial pattern formation during aggregation of the slime mould dictyostelium discoideum. J. Theor. Biol. 181, 203 (1996).","journal-title":"J. Theor. Biol."},{"key":"52344_CR6","doi-asserted-by":"publisher","first-page":"627","DOI":"10.1074\/jbc.271.2.627","volume":"271","author":"T Mair","year":"1996","unstructured":"Mair, T. & M\u00fcller, S. C. Traveling NADH and proton waves during oscillatory glycolysis in vitro. J. Biol. Chem. 271, 627 (1996).","journal-title":"J. Biol. Chem."},{"key":"52344_CR7","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.108.044205","volume":"108","author":"X Wang","year":"2023","unstructured":"Wang, X. et al. Composite spiral waves in discrete-time systems. Phys. Rev. E 108, 044205 (2023).","journal-title":"Phys. Rev. E"},{"key":"52344_CR8","first-page":"2516","volume":"79","author":"WSC Gurney","year":"1998","unstructured":"Gurney, W. S. C., Veitch, A. R., Cruickshank, I. & McGeachin, G. Circles and spirals: Population persistence in a spatially explicit predator-prey model. Ecology 79, 2516\u20132530 (1998).","journal-title":"Ecology"},{"issue":"3","key":"52344_CR9","doi-asserted-by":"publisher","first-page":"30005","DOI":"10.1209\/0295-5075\/ac2c98","volume":"136","author":"J Gao","year":"2022","unstructured":"Gao, J., Gu, C., Shen, C. & Yang, H. Spiral waves in population density distributions of invasive pests in warm-temperate deciduous forest ecosystems. EPL 136(3), 30005 (2022).","journal-title":"EPL"},{"issue":"100","key":"52344_CR10","doi-asserted-by":"publisher","first-page":"20140735","DOI":"10.1098\/rsif.2014.0735","volume":"11","author":"A Szolnoki","year":"2014","unstructured":"Szolnoki, A. et al. Cyclic dominance in evolutionary games: A review. J. R. Soc. Interface 11(100), 20140735 (2014).","journal-title":"J. R. Soc. Interface"},{"key":"52344_CR11","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.82.026107","volume":"82","author":"Y Qian","year":"2010","unstructured":"Qian, Y. et al. Diverse self-sustained oscillatory patterns and their mechanisms in excitable small-world networks. Phys. Rev. E 82, 026107 (2010).","journal-title":"Phys. Rev. E"},{"key":"52344_CR12","doi-asserted-by":"publisher","first-page":"1103","DOI":"10.1103\/PhysRevLett.71.1103","volume":"71","author":"A Karma","year":"1993","unstructured":"Karma, A. Spiral breakup in model equations of action potential propagation in cardiac tissue. Phys. Rev. Lett. 71, 1103 (1993).","journal-title":"Phys. Rev. Lett."},{"key":"52344_CR13","first-page":"H1088","volume":"291","author":"KH Tusscher","year":"2006","unstructured":"Tusscher, K. H. & Panfilov, A. V. Alternans and spiral breakup in a human ventricular tissue model. Am. J. Physiol. 291, H1088 (2006).","journal-title":"Am. J. Physiol."},{"key":"52344_CR14","doi-asserted-by":"publisher","first-page":"1599","DOI":"10.1038\/nn.2973","volume":"14","author":"J Viventi","year":"2011","unstructured":"Viventi, J. et al. Flexible, foldable, actively multiplexed, high-density electrode array for mapping brain activity in vivo. Nat. Neurosci. 14, 1599 (2011).","journal-title":"Nat. Neurosci."},{"key":"52344_CR15","doi-asserted-by":"publisher","DOI":"10.1016\/j.ecoleng.2019.105673","volume":"143","author":"J Gao","year":"2020","unstructured":"Gao, J. et al. Prediction of spatial distribution of invasive alien pests in two-dimensional systems based on a discrete time model. Ecol. Eng. 143, 105673 (2020).","journal-title":"Ecol. Eng."},{"key":"52344_CR16","doi-asserted-by":"publisher","first-page":"2494","DOI":"10.1103\/PhysRevLett.72.2494","volume":"72","author":"A Hagberg","year":"1994","unstructured":"Hagberg, A. & Meron, E. From labyrinthine patterns to spiral turbulence. Phys. Rev. Lett. 72, 2494 (1994).","journal-title":"Phys. Rev. Lett."},{"key":"52344_CR17","doi-asserted-by":"publisher","first-page":"477","DOI":"10.1063\/1.166047","volume":"4","author":"A Hagberg","year":"1994","unstructured":"Hagberg, A. & Meron, E. Complex patterns in reaction-diffusion systems: A tale of two front instabilities. Chaos 4, 477 (1994).","journal-title":"Chaos"},{"key":"52344_CR18","doi-asserted-by":"publisher","first-page":"1819","DOI":"10.1103\/PhysRevLett.78.1819","volume":"78","author":"AF Maree","year":"1997","unstructured":"Maree, A. F. & Panfilov, A. V. Spiral breakup in excitable tissue due to lateral instability. Phys. Rev. Lett. 78, 1819 (1997).","journal-title":"Phys. Rev. Lett."},{"key":"52344_CR19","doi-asserted-by":"publisher","first-page":"2811","DOI":"10.1103\/PhysRevLett.81.2811","volume":"81","author":"VS Zykov","year":"1998","unstructured":"Zykov, V. S., Mikhailov, A. S. & M\u00fcller, S. C. Wave instabilities in excitable media with fast inhibitor diffusion. Phys. Rev. Lett. 81, 2811 (1998).","journal-title":"Phys. Rev. Lett."},{"issue":"6","key":"52344_CR20","doi-asserted-by":"publisher","first-page":"1160","DOI":"10.1103\/PhysRevLett.82.1160","volume":"82","author":"M B\u00e4r","year":"1999","unstructured":"B\u00e4r, M. & Or-Guil, M. Alternative scenarios of spiral breakup in a reaction-diffusion model with excitable and oscillatory dynamics. Phys. Rev. Lett. 82(6), 1160 (1999).","journal-title":"Phys. Rev. Lett."},{"issue":"1","key":"52344_CR21","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0370-1573(92)90098-K","volume":"218","author":"E Meron","year":"1992","unstructured":"Meron, E. Pattern formation in excitable media. Phys. Rep. 218(1), 1\u201366 (1992).","journal-title":"Phys. Rep."},{"issue":"1","key":"52344_CR22","doi-asserted-by":"publisher","first-page":"164","DOI":"10.1103\/PhysRevLett.72.164","volume":"72","author":"D Barkley","year":"1994","unstructured":"Barkley, D. Euclidean symmetry and the dynamics of rotating spiral waves. Phys. Rev. Lett. 72(1), 164 (1994).","journal-title":"Phys. Rev. Lett."},{"issue":"13","key":"52344_CR23","doi-asserted-by":"publisher","first-page":"2090","DOI":"10.1103\/PhysRevLett.68.2090","volume":"68","author":"D Barkley","year":"1992","unstructured":"Barkley, D. Linear stability analysis of rotating spiral waves in excitable media. Phys. Rev. Lett. 68(13), 2090 (1992).","journal-title":"Phys. Rev. Lett."},{"key":"52344_CR24","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1137\/S0036139993259337","volume":"56","author":"M Courtemanche","year":"1996","unstructured":"Courtemanche, M., Keener, J. P. & Glass, L. A delay equation representation of pulse circulation on a ring in excitable media. SIAM J. Appl. Math. 56, 119 (1996).","journal-title":"SIAM J. Appl. Math."},{"key":"52344_CR25","doi-asserted-by":"publisher","first-page":"772","DOI":"10.1126\/science.1169973","volume":"324","author":"J Horv\u00e1th","year":"2009","unstructured":"Horv\u00e1th, J., Szalai, I. & Kepper, P. An experimental design method leading to chemical Turing patterns. Science 324, 772\u2013775 (2009).","journal-title":"Science"},{"key":"52344_CR26","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.107.014216","volume":"107","author":"S Gao","year":"2023","unstructured":"Gao, S., Chang, L., Perc, M. & Wang, Z. Turing patterns in simplicial complexes. Phys. Rev. E 107, 014216 (2023).","journal-title":"Phys. Rev. E"},{"key":"52344_CR27","doi-asserted-by":"publisher","first-page":"544","DOI":"10.1038\/nphys1651","volume":"6","author":"H Nakao","year":"2010","unstructured":"Nakao, H. & Mikhailov, A. Turing patterns in network-organized activator\u2013inhibitor systems. Nat. Phys. 6, 544\u2013550 (2010).","journal-title":"Nat. Phys."},{"key":"52344_CR28","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.86.056203","volume":"86","author":"L Fernandes","year":"2012","unstructured":"Fernandes, L. & Aguiar, M. Turing patterns and apparent competition in predator-prey food webs on networks. Phys. Rev. E 86, 056203 (2012).","journal-title":"Phys. Rev. E"},{"key":"52344_CR29","doi-asserted-by":"publisher","first-page":"6572","DOI":"10.1073\/pnas.1720770115","volume":"115","author":"D Karig","year":"2018","unstructured":"Karig, D., Martini, K. M., Lu, T. & Weiss, R. Stochastic Turing patterns in a synthetic bacterial population. Proc. Natl. Acad. Sci. U.S.A. 115, 6572\u20136577 (2018).","journal-title":"Proc. Natl. Acad. Sci. U.S.A."},{"key":"52344_CR30","doi-asserted-by":"publisher","first-page":"20140218","DOI":"10.1098\/rstb.2014.0218","volume":"370","author":"P Ball","year":"2015","unstructured":"Ball, P. Forging patterns and making waves from biology to geology: A commentary on Turing (1952) \u2018The chemical basis of morphogenesis\u2019. Philos. T. R. Soc. B 370, 20140218\u201320140218 (2015).","journal-title":"Philos. T. R. Soc. B"},{"key":"52344_CR31","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.99.012407","volume":"99","author":"D Vidmar","year":"2019","unstructured":"Vidmar, D. & Rappel, W. J. Extinction dynamics of spiral defect chaos. Phys. Rev. E 99, 012407 (2019).","journal-title":"Phys. Rev. E"}],"container-title":["Scientific Reports"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.nature.com\/articles\/s41598-026-52344-7","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s41598-026-52344-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s41598-026-52344-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T22:25:35Z","timestamp":1783981535000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.nature.com\/articles\/s41598-026-52344-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5,13]]},"references-count":31,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,12]]}},"alternative-id":["52344"],"URL":"https:\/\/doi.org\/10.1038\/s41598-026-52344-7","relation":{},"ISSN":["2045-2322"],"issn-type":[{"value":"2045-2322","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,5,13]]},"assertion":[{"value":"30 November 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 May 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 May 2026","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 declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"21871"}}