{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T16:03:39Z","timestamp":1772294619472,"version":"3.50.1"},"reference-count":8,"publisher":"World Scientific Pub Co Pte Lt","issue":"07","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2014,7]]},"abstract":"<jats:p> Chaos in physical systems governed by nonlinear PDEs have been amply observed and reported. However, rigorous proofs for their occurrences are challenging. In particular, for a second order PDE of hyperbolic type with a van der Pol cubic nonlinearity in one of the boundary conditions and a spatially distributed antidamping term in a linear governing equation, no proof for the onset of chaos was available even though chaos was expected to occur when the antidamping term becomes sufficiently strong. In this paper, we use an operator-factoring technique together with the analogy with the one-dimensional wave equation to prove that for the Klein\u2013Gordon equation chaos occurs for a class of equations and boundary conditions when system parameters enter a certain regime. Chaotic and nonchaotic profiles of solutions are illustrated by computer graphics. <\/jats:p>","DOI":"10.1142\/s0218127414300213","type":"journal-article","created":{"date-parts":[[2014,7,30]],"date-time":"2014-07-30T05:32:54Z","timestamp":1406698374000},"page":"1430021","source":"Crossref","is-referenced-by-count":16,"title":["Chaotic Oscillations of Solutions of the Klein\u2013Gordon Equation Due to Imbalance of Distributed and Boundary Energy Flows"],"prefix":"10.1142","volume":"24","author":[{"given":"Goong","family":"Chen","sequence":"first","affiliation":[{"name":"Institute for Quantum Science and Engineering (IQSE), Department of Mathematics, Texas A&amp;M University, College Station, TX 77843, USA"},{"name":"Science Program, Texas A&amp;M University at Qatar, Education City, Doha, Qatar"}]},{"given":"Bo","family":"Sun","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Changsha University of Science and Technology, Changsha, Hunan, P. R. China"},{"name":"Department of Mathematics, Texas A&amp;M University, College Station, TX 77843, USA"}]},{"given":"Tingwen","family":"Huang","sequence":"additional","affiliation":[{"name":"Science Program, Texas A&amp;M University at Qatar, Education City, Doha, Qatar"}]}],"member":"219","published-online":{"date-parts":[[2014,7,30]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-98-02022-4"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127498000280"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127498000292"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1063\/1.532670"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127411029136"},{"key":"rf6","volume-title":"An Introduction to Chaotic Dynamical Systems","author":"Devaney R. L.","year":"2003"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2005.10.019"},{"key":"rf8","series-title":"Notas de Matem\u00e1tica","volume-title":"The Energy Method in Nonlinear Partial Differential Equations","author":"Strauss W. A.","year":"1969"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127414300213","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T00:59:25Z","timestamp":1565139565000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127414300213"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,7]]},"references-count":8,"journal-issue":{"issue":"07","published-online":{"date-parts":[[2014,7,30]]},"published-print":{"date-parts":[[2014,7]]}},"alternative-id":["10.1142\/S0218127414300213"],"URL":"https:\/\/doi.org\/10.1142\/s0218127414300213","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,7]]}}}