{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,28]],"date-time":"2026-09-28T18:37:41Z","timestamp":1790620661269,"version":"4.1.0"},"reference-count":34,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2023,9,21]],"date-time":"2023-09-21T00:00:00Z","timestamp":1695254400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Stud Appl Math"],"published-print":{"date-parts":[[2024,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>It is known that rank\u2010two bimolecular mass\u2010action systems do not admit limit cycles. With a view to understanding which small mass\u2010action systems admit oscillation, in this paper we study rank\u2010two networks with bimolecular source complexes but allow target complexes with higher molecularities. As our goal is to find oscillatory networks of minimal size, we focus on networks with three reactions, the minimum number that is required for oscillation. However, some of our intermediate results are valid in greater\u00a0generality.<\/jats:p>\n                  <jats:p>One key finding is that an isolated periodic orbit cannot occur in a three\u2010reaction, trimolecular, mass\u2010action system with bimolecular sources. In fact, we characterize all networks in this class that admit a periodic orbit; in every case, all nearby orbits are periodic too. Apart from the well\u2010known Lotka and Ivanova reactions, we identify another network in this class that admits a center. This new network exhibits a vertical Andronov\u2013Hopf\u00a0bifurcation.<\/jats:p>\n                  <jats:p>Furthermore, we characterize all two\u2010species, three\u2010reaction, bimolecular\u2010sourced networks that admit an Andronov\u2013Hopf bifurcation with mass\u2010action kinetics. These include two families of networks that admit a supercritical Andronov\u2013Hopf bifurcation and hence a stable limit cycle. These networks necessarily have a target complex with a molecularity of at least four, and it turns out that there are exactly four such networks that are\u00a0tetramolecular.<\/jats:p>","DOI":"10.1111\/sapm.12639","type":"journal-article","created":{"date-parts":[[2023,9,21]],"date-time":"2023-09-21T09:45:43Z","timestamp":1695289543000},"page":"249-278","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Oscillations in three\u2010reaction quadratic mass\u2010action systems"],"prefix":"10.1111","volume":"152","author":[{"given":"Murad","family":"Banaji","sequence":"first","affiliation":[{"name":"Mathematical Institute University of Oxford  Oxford UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bal\u00e1zs","family":"Boros","sequence":"additional","affiliation":[{"name":"Department of Mathematics University of Vienna  Vienna Austria"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Josef","family":"Hofbauer","sequence":"additional","affiliation":[{"name":"Department of Mathematics University of Vienna  Vienna Austria"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2023,9,21]]},"reference":[{"key":"e_1_2_9_2_1","doi-asserted-by":"publisher","DOI":"10.1021\/ja01453a010"},{"key":"e_1_2_9_3_1","volume-title":"Reaction kinetics: Exercises, programs and theorems","author":"T\u00f3th J","year":"2018"},{"key":"e_1_2_9_4_1","volume-title":"Analysis in classes of discontinuous functions and equations of mathematical physics","author":"Vol'pert AI","year":"1985"},{"key":"e_1_2_9_5_1","doi-asserted-by":"publisher","DOI":"10.1063\/1.444861"},{"key":"e_1_2_9_6_1","doi-asserted-by":"publisher","DOI":"10.1039\/F29858100115"},{"key":"e_1_2_9_7_1","doi-asserted-by":"publisher","DOI":"10.14232\/ejqtde.2022.1.42"},{"key":"e_1_2_9_8_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01165134"},{"key":"e_1_2_9_9_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.nonrwa.2023.103839"},{"key":"e_1_2_9_10_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2017.12.012"},{"key":"e_1_2_9_11_1","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6544\/acb0a8"},{"key":"e_1_2_9_12_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1432-1033.1968.tb00175.x"},{"key":"e_1_2_9_13_1","doi-asserted-by":"publisher","DOI":"10.1016\/0022-5193(71)90054-3"},{"key":"e_1_2_9_14_1","doi-asserted-by":"publisher","DOI":"10.1016\/0022-5193(79)90042-0"},{"issue":"1","key":"e_1_2_9_15_1","first-page":"79","article-title":"On the possibility of auto\u2010oscillation in homogeneous chemical systems with quadratic autocatalysis","volume":"17","author":"Frank\u2010Kamenetsky DA","year":"1943","journal-title":"Zh Fiz Khim"},{"key":"e_1_2_9_16_1","doi-asserted-by":"publisher","DOI":"10.1039\/F29858101487"},{"key":"e_1_2_9_17_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01166095"},{"issue":"1","key":"e_1_2_9_18_1","first-page":"153","article-title":"Atoms of multistationarity in chemical reaction networks","volume":"51","author":"Joshi B","year":"2013","journal-title":"J Math Chem"},{"issue":"1598","key":"e_1_2_9_19_1","first-page":"331","article-title":"Stability and complex balancing in mass\u2010action systems with three short complexes","volume":"334","author":"Horn F.","year":"1973","journal-title":"Proc R Soc A"},{"key":"e_1_2_9_20_1","doi-asserted-by":"publisher","DOI":"10.1137\/22M1478392"},{"key":"e_1_2_9_21_1","doi-asserted-by":"publisher","DOI":"10.1002\/ijch.201800003"},{"key":"e_1_2_9_22_1","doi-asserted-by":"publisher","DOI":"10.1137\/17M1129076"},{"key":"e_1_2_9_23_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01458147"},{"key":"e_1_2_9_24_1","unstructured":"BanajiM.Counting chemical reaction networks with NAUTY.2017.https:\/\/arxiv.org\/pdf\/1705.10820.pdf"},{"key":"e_1_2_9_25_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.mbs.2007.07.003"},{"key":"e_1_2_9_26_1","doi-asserted-by":"publisher","DOI":"10.1016\/0009-2509(87)80099-4"},{"key":"e_1_2_9_27_1","doi-asserted-by":"publisher","DOI":"10.1137\/15M1034441"},{"key":"e_1_2_9_28_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4211-4"},{"key":"e_1_2_9_29_1","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(78)90121-3"},{"key":"e_1_2_9_30_1","doi-asserted-by":"publisher","DOI":"10.1216\/rmjm\/1181072288"},{"key":"e_1_2_9_31_1","unstructured":"BorosB HofbauerJ.Oscillations in planar deficiency\u2010one mass\u2010action systems.J Dyn Diff Equat. Published online August 9 2021. doi:https:\/\/doi.org\/10.1007\/s10884\u2010021\u201010051\u2010z"},{"key":"e_1_2_9_32_1","unstructured":"BorosB.Reaction networks GitHub repository.https:\/\/github.com\/balazsboros\/reaction_networks"},{"key":"e_1_2_9_33_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3978-7"},{"key":"e_1_2_9_34_1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139173179"},{"key":"e_1_2_9_35_1","first-page":"127\u00a0109","article-title":"Adding species to chemical reaction networks: Preserving rank preserves nondegenerate behaviours","volume":"426","author":"Banaji M","year":"2022","journal-title":"Appl Math Comput"}],"container-title":["Studies in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1111\/sapm.12639","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,3,21]],"date-time":"2024-03-21T06:26:22Z","timestamp":1711002382000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1111\/sapm.12639"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,21]]},"references-count":34,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,1]]}},"alternative-id":["10.1111\/sapm.12639"],"URL":"https:\/\/doi.org\/10.1111\/sapm.12639","archive":["Portico"],"relation":{},"ISSN":["0022-2526","1467-9590"],"issn-type":[{"value":"0022-2526","type":"print"},{"value":"1467-9590","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,21]]},"assertion":[{"value":"2023-04-08","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2023-08-22","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2023-09-21","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}