{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:01:38Z","timestamp":1760148098183,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T00:00:00Z","timestamp":1680220800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Spanish Agencia Estatal de Investigaci\u00f3n","award":["MCI-21-PID2020-113052GB-I00","PID2019-104658GB-I00","MSCA-RISE-2017-777911","2022-SGR 00113"],"award-info":[{"award-number":["MCI-21-PID2020-113052GB-I00","PID2019-104658GB-I00","MSCA-RISE-2017-777911","2022-SGR 00113"]}]},{"name":"Agencia Estatal de Investigaci\u00f3n","award":["MCI-21-PID2020-113052GB-I00","PID2019-104658GB-I00","MSCA-RISE-2017-777911","2022-SGR 00113"],"award-info":[{"award-number":["MCI-21-PID2020-113052GB-I00","PID2019-104658GB-I00","MSCA-RISE-2017-777911","2022-SGR 00113"]}]},{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["MCI-21-PID2020-113052GB-I00","PID2019-104658GB-I00","MSCA-RISE-2017-777911","2022-SGR 00113"],"award-info":[{"award-number":["MCI-21-PID2020-113052GB-I00","PID2019-104658GB-I00","MSCA-RISE-2017-777911","2022-SGR 00113"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]},{"name":"AGAUR (Generalitat de Catalunya)","award":["MCI-21-PID2020-113052GB-I00","PID2019-104658GB-I00","MSCA-RISE-2017-777911","2022-SGR 00113"],"award-info":[{"award-number":["MCI-21-PID2020-113052GB-I00","PID2019-104658GB-I00","MSCA-RISE-2017-777911","2022-SGR 00113"]}]},{"name":"Acad\u00e8mia de Ci\u00e8ncies i Arts de Barcelona","award":["MCI-21-PID2020-113052GB-I00","PID2019-104658GB-I00","MSCA-RISE-2017-777911","2022-SGR 00113"],"award-info":[{"award-number":["MCI-21-PID2020-113052GB-I00","PID2019-104658GB-I00","MSCA-RISE-2017-777911","2022-SGR 00113"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we study how many algebraic limit cycles have the discontinuous piecewise linear differential systems separated by a straight line, with polynomial first integrals on both sides. We assume that at least one of the systems is Hamiltonian. Under this assumption, piecewise differential systems have no more than one limit cycle. This study characterizes linear differential systems with polynomial first integrals.<\/jats:p>","DOI":"10.3390\/axioms12040342","type":"journal-article","created":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T08:27:27Z","timestamp":1680251247000},"page":"342","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Limit Cycles of Polynomially Integrable Piecewise Differential Systems"],"prefix":"10.3390","volume":"12","author":[{"given":"Bel\u00e9n","family":"Garc\u00eda","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad de Oviedo, Avda Calvo Sotelo s\/n, 33007 Oviedo, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9511-5999","authenticated-orcid":false,"given":"Jaume","family":"Llibre","sequence":"additional","affiliation":[{"name":"Departament de Matem\u00e0tiques, Universitat Aut\u00f2noma de Barcelona, Bellaterra, 08193 Barcelona, Spain"}]},{"given":"Jes\u00fas S.","family":"P\u00e9rez del R\u00edo","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad de Oviedo, Avda Calvo Sotelo s\/n, 33007 Oviedo, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1522-7086","authenticated-orcid":false,"given":"Set","family":"P\u00e9rez-Gonz\u00e1lez","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad de Oviedo, Avda Calvo Sotelo s\/n, 33007 Oviedo, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Filippov, A.F. (1988). Differential Equations with Discontinuous Right\u2013Hand Sides, Kluwer Academic Publishers Group.","DOI":"10.1007\/978-94-015-7793-9"},{"key":"ref_2","unstructured":"Andronov, A., Vitt, A., and Khaikin, S. (1966). Theory of Oscillations, Pergamon Press."},{"key":"ref_3","unstructured":"Di Bernardo, M., Budd, C.J., Champneys, A.R., and Kowalczyk, P. (2008). Piecewise-Smooth Dynamical Systems: Theory and Applications, Springer."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Simpson, D.J.W. (2010). Bifurcations in Piecewise-Smooth Continuous Systems, World Scientific.","DOI":"10.1142\/7612"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1826","DOI":"10.1016\/j.physd.2012.08.002","article-title":"Dynamics and bifurcations of nonsmooth systems: A survey","volume":"241","author":"Makarenko","year":"2012","journal-title":"Physica D"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1016\/j.jmaa.2014.10.077","article-title":"On the number of limit cycles in discontinuous piecewise linear differential systems with two pieces separated by a straight line","volume":"424","author":"Llibre","year":"2015","journal-title":"J. Math. Anal. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1850039","DOI":"10.1142\/S0218127418500396","article-title":"Algebraic limit cycles in piecewise linear differential systems","volume":"28","author":"Buzzi","year":"2018","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"5416","DOI":"10.1016\/j.na.2012.04.043","article-title":"Quadratic systems with an integrable saddle: A complete classification in the coefficient space R12","volume":"75","author":"Llibre","year":"2012","journal-title":"Nonlinear Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1016\/j.jde.2022.11.007","article-title":"Limit cycles in piecewise polynomial Hamiltonian systems allowing nonlinear switching boundaries","volume":"344","author":"Li","year":"2023","journal-title":"J. Differ. Equ."},{"key":"ref_10","first-page":"3091","article-title":"Limit cycles for regularized discontinuous dynamical systems with a hyperplane of discontinuity","volume":"22","author":"Dieci","year":"2017","journal-title":"Discret. Contin. Dyn. Syst. B"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/j.jde.2022.09.028","article-title":"The number of limit cycles for regularized piecewise polynomial systems is unbounded","volume":"342","author":"Huzak","year":"2023","journal-title":"J. Differ. Equ."},{"key":"ref_12","unstructured":"Sotomayor, J., and Teixeira, M.A. (1995, January 24\u201329). Regularization of discontinuous vector fields. Proceedings of the International Conference on Differential Equations, Lisboa, Portugal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1137\/11083928X","article-title":"Canonical Discontinuous Planar Piecewise Linear Systems","volume":"11","author":"Freire","year":"2012","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"420","DOI":"10.1007\/BF02874780","article-title":"Phase portraits of the quadratic vector fields with a polynomial first integral","volume":"55","author":"Llibre","year":"2006","journal-title":"Rend. Circ. Mat. Palermo"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/4\/342\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:07:22Z","timestamp":1760123242000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/4\/342"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,31]]},"references-count":14,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["axioms12040342"],"URL":"https:\/\/doi.org\/10.3390\/axioms12040342","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2023,3,31]]}}}