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For some diffeomorphisms with the expansion dimension equal to one or two in three-dimensional spaces, the conditions under which there exist Smale horseshoes and uniformly hyperbolic invariant sets on which the systems are topologically conjugate to the two-sided fullshift on finite alphabet are obtained; for some expanding maps, the chaotic region is analyzed by using the coupled-expansion theory and the Brouwer degree theory. For three types of higher-dimensional polynomial maps with degree two, the conditions under which there are Smale horseshoes and uniformly hyperbolic invariant sets are given, and the topological conjugacy between the maps on the invariant sets and the two-sided fullshift on finite alphabet is obtained. Some interesting maps with chaotic attractors and positive Lyapunov exponents in three-dimensional spaces are found by using computer simulations. In the end, two examples are provided to illustrate the theoretical results.<\/jats:p>","DOI":"10.1142\/s0218127416501315","type":"journal-article","created":{"date-parts":[[2016,8,5]],"date-time":"2016-08-05T09:03:24Z","timestamp":1470387804000},"page":"1650131","source":"Crossref","is-referenced-by-count":14,"title":["Chaotic Polynomial Maps"],"prefix":"10.1142","volume":"26","author":[{"given":"Xu","family":"Zhang","sequence":"first","affiliation":[{"name":"Department of Mathematics, Shandong University, Weihai, Shandong 264209, P. R. 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