{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:06:52Z","timestamp":1760058412907,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,3,28]],"date-time":"2025-03-28T00:00:00Z","timestamp":1743120000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Key Scientific and Technological Research Project of Science and Technology Department of Zhanjiang City","award":["240418034547512","2018QZJ03","2018RZJ03","2024RC057"],"award-info":[{"award-number":["240418034547512","2018QZJ03","2018RZJ03","2024RC057"]}]},{"name":"Opening Project of Bridge Non-destruction Detecting and Engineering Computing Key Laboratory of Sichuan Province","award":["240418034547512","2018QZJ03","2018RZJ03","2024RC057"],"award-info":[{"award-number":["240418034547512","2018QZJ03","2018RZJ03","2024RC057"]}]},{"name":"Opening Project of Artificial Intelligence Key Laboratory of Sichuan Province","award":["240418034547512","2018QZJ03","2018RZJ03","2024RC057"],"award-info":[{"award-number":["240418034547512","2018QZJ03","2018RZJ03","2024RC057"]}]},{"name":"Scientic Research Project of SUSE","award":["240418034547512","2018QZJ03","2018RZJ03","2024RC057"],"award-info":[{"award-number":["240418034547512","2018QZJ03","2018RZJ03","2024RC057"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we discussed the cofinite sensitivity, shadowing property (SP), P-chaos, and chain mixing of a system induced by symmetric maps (Cournot maps) D(a,b)=(t(b),s(a)) over a product space G\u00d7H, where s:G\u2192H, t:H\u2192G, a\u2208G, b\u2208H, G and H are closed subintervals with G,H\u2282R. The following hold: (1) D is cofinitely sensitive and equivalent to D2|\u03931 or D2|\u03932 being sensitive, where \u03931={(t(b),b):b\u2208H}, \u03932={(a,s(a)):a\u2208G}. (2) D possessing an SP is equivalent to both s\u2218t and t\u2218s having an SP. (3) t\u2218s possesses an SP if and only if s\u2218t does as well. (4) D is P-chaotic and equivalent to the maps s\u2218t and t\u2218s being P-chaotic. (5) If D is chain mixing, then both D2|\u03931 and D2|\u03932 are chain mixing. (6) If D2|\u03931 and D2|\u03932 are chain mixing, then D is chain transitive. Moreover, we extended (1)\u2013(4) to three-dimensional cases.<\/jats:p>","DOI":"10.3390\/sym17040511","type":"journal-article","created":{"date-parts":[[2025,3,28]],"date-time":"2025-03-28T06:06:35Z","timestamp":1743141995000},"page":"511","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Sensitivity, Shadowing Property and P-Chaos in Duopoly Games"],"prefix":"10.3390","volume":"17","author":[{"given":"Hongqing","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematics and Computer Science, Guangdong Ocean University, Zhanjiang 524025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8806-9656","authenticated-orcid":false,"given":"Tianxiu","family":"Lu","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3381-5786","authenticated-orcid":false,"given":"Risong","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Guangdong Ocean University, Zhanjiang 524025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ping","family":"Gao","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, China"},{"name":"Bridge Non-destruction Detecting and Engineering Computing Key Laboratory of Sichuan Province, Zigong 643000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Puu, T., and Sushko, I. 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