{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T09:19:27Z","timestamp":1740129567549,"version":"3.37.3"},"reference-count":28,"publisher":"IOP Publishing","issue":"8","license":[{"start":{"date-parts":[[2021,7,6]],"date-time":"2021-07-06T00:00:00Z","timestamp":1625529600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"},{"start":{"date-parts":[[2021,7,6]],"date-time":"2021-07-06T00:00:00Z","timestamp":1625529600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/iopscience.iop.org\/info\/page\/text-and-data-mining"}],"funder":[{"name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","award":["UID\/MAT\/00144\/2019"],"award-info":[{"award-number":["UID\/MAT\/00144\/2019"]}]},{"DOI":"10.13039\/501100004281","name":"Narodowe Centrum Nauki","doi-asserted-by":"crossref","award":["2016\/23\/G\/ST1\/04081"],"award-info":[{"award-number":["2016\/23\/G\/ST1\/04081"]}],"id":[{"id":"10.13039\/501100004281","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"short-container-title":["Nonlinearity"],"published-print":{"date-parts":[[2021,8,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Coupled cell systems associated with a coupled cell network are determined by (smooth) vector fields that are consistent with the network structure. Here, we follow the formalisms of Stewart <jats:italic>et\u00a0al<\/jats:italic> (2003 <jats:italic>SIAM J. Appl. Dyn. Syst.<\/jats:italic> \n                  <jats:bold>2<\/jats:bold> 609\u2013646), Golubitsky <jats:italic>et\u00a0al<\/jats:italic> (2005 <jats:italic>SIAM J. Appl. Dyn. Syst.<\/jats:italic> \n                  <jats:bold>4<\/jats:bold> 78\u2013100) and Field (2004 <jats:italic>Dyn. Syst.<\/jats:italic> \n                  <jats:bold>19<\/jats:bold> 217\u2013243). It is known that two non-isomorphic <jats:italic>n<\/jats:italic>-cell coupled networks can determine the same sets of vector fields\u2014these networks are said to be ordinary differential equation (ODE)-equivalent. The set of all <jats:italic>n<\/jats:italic>-cell coupled networks is so partitioned into classes of ODE-equivalent networks. With no further restrictions, the number of ODE-classes is not finite and each class has an infinite number of networks. Inside each ODE-class we can find a finite subclass of networks that minimize the number of edges in the class, called minimal networks. In this paper, we consider coupled cell networks with asymmetric inputs. That is, if <jats:italic>k<\/jats:italic> is the number of distinct edges types, these networks have the property that every cell receives <jats:italic>k<\/jats:italic> inputs, one of each type. Fixing the number <jats:italic>n<\/jats:italic> of cells, we prove that: the number of ODE-classes is finite; restricting to a maximum of <jats:italic>n<\/jats:italic>(<jats:italic>n<\/jats:italic> \u2212 1) inputs, we can cover all the ODE-classes; all minimal <jats:italic>n<\/jats:italic>-cell networks with <jats:italic>n<\/jats:italic>(<jats:italic>n<\/jats:italic> \u2212 1) asymmetric inputs are ODE-equivalent. We also give a simple criterion to test if a network is minimal and we conjecture lower estimates for the number of distinct ODE-classes of <jats:italic>n<\/jats:italic>-cell networks with any number <jats:italic>k<\/jats:italic> of asymmetric inputs. Moreover, we present a full list of representatives of the ODE-classes of networks with three cells and two asymmetric inputs.<\/jats:p>","DOI":"10.1088\/1361-6544\/ac0b2e","type":"journal-article","created":{"date-parts":[[2021,7,7]],"date-time":"2021-07-07T12:47:39Z","timestamp":1625662059000},"page":"5630-5661","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":2,"title":["Towards a classification of networks with asymmetric inputs"],"prefix":"10.1088","volume":"34","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3508-0509","authenticated-orcid":false,"given":"Manuela","family":"Aguiar","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8852-6175","authenticated-orcid":false,"given":"Ana","family":"Dias","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1869-7662","authenticated-orcid":false,"given":"Pedro","family":"Soares","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2021,7,6]]},"reference":[{"key":"nonac0b2ebib1","doi-asserted-by":"publisher","first-page":"1245","DOI":"10.1088\/0951-7715\/23\/6\/001","article-title":"Dynamical equivalence of networks of coupled dynamical systems: I. 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