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In this paper, we take advantage of a topologically dynamical method to characterize the dynamical behaviors of the systems, including sensitivity, transitivity and Li\u2013Yorke chaos. For this end, we consider a system [Formula: see text] induced by a sequence of continuous maps and its functional envelope [Formula: see text], and show that, under some considerable condition, [Formula: see text] is transitive if and only if [Formula: see text] is weakly mixing of order [Formula: see text]; [Formula: see text] is Li\u2013Yorke chaotic and sensitive if [Formula: see text] is strongly mixing. Those abstract results have their own significance and can be applied to such kind of equations. <\/jats:p>","DOI":"10.1142\/s0218127420502272","type":"journal-article","created":{"date-parts":[[2020,12,10]],"date-time":"2020-12-10T10:44:48Z","timestamp":1607597088000},"page":"2050227","source":"Crossref","is-referenced-by-count":0,"title":["Chaotic Behaviors of One-Dimensional Wave Equations with van der Pol Boundary Conditions Containing a Source Term"],"prefix":"10.1142","volume":"30","author":[{"given":"Zhijing","family":"Chen","sequence":"first","affiliation":[{"name":"School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou 510665, P. R. China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0148-1024","authenticated-orcid":false,"given":"Yu","family":"Huang","sequence":"additional","affiliation":[{"name":"School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P. R. China"}]},{"given":"Haiwei","family":"Sun","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Macau, Macao 999078, P. R. China"}]},{"given":"Tongyang","family":"Zhou","sequence":"additional","affiliation":[{"name":"Yantai Engineering and Technology College, Yantai 264000, P. R. China"}]}],"member":"219","published-online":{"date-parts":[[2020,12,9]]},"reference":[{"key":"S0218127420502272BIB001","series-title":"Contemp. Math.","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1090\/conm\/356\/06496","volume-title":"Chapel Hill Ergodic Theory Workshops","volume":"356","author":"Akin E.","year":"2004"},{"key":"S0218127420502272BIB002","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/j.aml.2012.08.014","volume":"26","author":"Arcila M. M.","year":"2013","journal-title":"Appl. Math. 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