{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:29:06Z","timestamp":1760149746322,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,9,7]],"date-time":"2023-09-07T00:00:00Z","timestamp":1694044800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Cooperative Education Project of the Ministry of Education","award":["220605115023220","2023NSFSC0070","2022NSFSC1821","2020RC24","2020QT13"],"award-info":[{"award-number":["220605115023220","2023NSFSC0070","2022NSFSC1821","2020RC24","2020QT13"]}]},{"name":"Natural Science Foundation of Sichuan Province","award":["220605115023220","2023NSFSC0070","2022NSFSC1821","2020RC24","2020QT13"],"award-info":[{"award-number":["220605115023220","2023NSFSC0070","2022NSFSC1821","2020RC24","2020QT13"]}]},{"name":"Scientific Research Project of SUSE","award":["220605115023220","2023NSFSC0070","2022NSFSC1821","2020RC24","2020QT13"],"award-info":[{"award-number":["220605115023220","2023NSFSC0070","2022NSFSC1821","2020RC24","2020QT13"]}]},{"name":"Ministry of Education Science and Technology Development Center","award":["220605115023220","2023NSFSC0070","2022NSFSC1821","2020RC24","2020QT13"],"award-info":[{"award-number":["220605115023220","2023NSFSC0070","2022NSFSC1821","2020RC24","2020QT13"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Chaos is a common phenomenon in nature and social sciences. As is well known, chaos has multiple definitions, and there are both differences and connections between them. The unique properties of chaotic systems can be leveraged to address challenges in communication, security, data processing, system analysis, and control across different domains. For semi-flows, this paper introduces two important concepts corresponding to discrete dynamical systems, finitely chaotic and pairwise sensitivity. Since Tent map and its induced suspended semi-flows both have these two properties, then these two concepts on the semi-flows have extensive and important applications and meanings in information security, finance, artificial intelligence and other fields. This paper extends the vast majority of corresponding results in discrete dynamical systems to semi-flows.<\/jats:p>","DOI":"10.3390\/axioms12090860","type":"journal-article","created":{"date-parts":[[2023,9,7]],"date-time":"2023-09-07T10:11:07Z","timestamp":1694081467000},"page":"860","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Finite Chaoticity and Pairwise Sensitivity of a Strong-Mixing Measure-Preserving Semi-Flow"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3381-5786","authenticated-orcid":false,"given":"Risong","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematics and Computer Science, Guangdong Ocean University, Zhanjiang 524025, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2760-9866","authenticated-orcid":false,"given":"Jingmin","family":"Pi","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, China"}]},{"given":"Yongjiang","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Guangdong Ocean University, Zhanjiang 524025, China"}]},{"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"}]},{"given":"Jianjun","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sichuan Agricultural University, Yaan 625014, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0275-1292","authenticated-orcid":false,"given":"Xianfeng","family":"Ding","sequence":"additional","affiliation":[{"name":"School of Science, Southwest Petroleum University, Chengdu 610500, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1090\/S0002-9939-1983-0677230-7","article-title":"A chaotic function with some extremal properties","volume":"87","year":"1983","journal-title":"Proc. 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