{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T03:40:59Z","timestamp":1776397259117,"version":"3.51.2"},"reference-count":13,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2025,3,10]],"date-time":"2025-03-10T00:00:00Z","timestamp":1741564800000},"content-version":"vor","delay-in-days":68,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2025,1]]},"abstract":"<jats:p>This article extends Alfredo Peris\u2019s work on chaos in set\u2010valued dynamics by providing new characterizations and applications of transitivity and mixing properties. Peris demonstrated that the topological transitivity of a set\u2010valued map is closely related to the weak mixing property of the individual map. In this continuation, we investigate these relationships in a broader context, including general metric spaces and infinite\u2010dimensional Banach spaces. First, we extend the characterization of transitivity and mixing in set\u2010valued dynamic systems by exploring additional conditions in general metric spaces. Then, we apply these results to the dynamics of fractal sets, showing how these properties influence the structure and behavior of chaotic attractors and Julia sets. Additionally, we incorporate numerical simulations and visualizations to illustrate the theoretical concepts and demonstrate examples of chaotic behavior in set\u2010valued systems. These simulations provide a visual and computational tool to better understand the dynamics of set\u2010valued maps, making abstract theories more accessible and engaging. Finally, we address linear dynamics in infinite\u2010dimensional Banach spaces, providing new proofs and characterizations that relate the hypercyclicity criterion with the transitivity and mixing properties in linear operators. This work not only expands the existing theoretical results but also offers new perspectives and tools for studying complex dynamic systems with potential applications in mathematics and physics.<\/jats:p>","DOI":"10.1155\/jama\/4134128","type":"journal-article","created":{"date-parts":[[2025,3,12]],"date-time":"2025-03-12T03:31:58Z","timestamp":1741750318000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Advanced Extensions and Applications of Transitivity and Mixing in Set\u2010Valued Dynamics With Numerical Simulations and Visual Insights"],"prefix":"10.1155","volume":"2025","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9647-6163","authenticated-orcid":false,"given":"Illych","family":"Alvarez","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2025,3,10]]},"reference":[{"key":"e_1_2_8_1_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2004.12.039"},{"key":"e_1_2_8_2_2","doi-asserted-by":"publisher","DOI":"10.1080\/00029890.1975.11994008"},{"key":"e_1_2_8_3_2","doi-asserted-by":"publisher","DOI":"10.3390\/math9202629"},{"key":"e_1_2_8_4_2","first-page":"3109","article-title":"Recent progress in dynamics of multi-valued maps","volume":"142","author":"Balibrea F.","year":"2014","journal-title":"Proceedings of the American Mathematical Society"},{"key":"e_1_2_8_5_2","first-page":"3699","article-title":"Chaotic polynomial dynamics","volume":"20","author":"Berm\u00fadez T.","year":"2010","journal-title":"International Journal of Bifurcation and Chaos"},{"key":"e_1_2_8_6_2","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/015"},{"key":"e_1_2_8_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/0166-8641(82)90065-7"},{"key":"e_1_2_8_8_2","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-1992-00328-2"},{"key":"e_1_2_8_9_2","volume-title":"Topological and Symbolic Dynamics","author":"Kurka P.","year":"2003"},{"key":"e_1_2_8_10_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-70335-5"},{"key":"e_1_2_8_11_2","volume-title":"Ergodic Theory","author":"Petersen K.","year":"1989"},{"key":"e_1_2_8_12_2","first-page":"453","article-title":"A note on topologically weakly mixing homeomorphisms","volume":"9","author":"Ward T.","year":"1989","journal-title":"Ergodic Theory and Dynamical Systems"},{"key":"e_1_2_8_13_2","doi-asserted-by":"crossref","unstructured":"AlvarezI. Advanced extensions and applications of transitivity and mixing in set-valued dynamics with numerical simulations and visual insights 2024 https:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract_id=4913534.","DOI":"10.2139\/ssrn.4913534"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/jama\/4134128","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1155\/jama\/4134128","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/jama\/4134128","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T02:42:13Z","timestamp":1776393733000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/jama\/4134128"}},"subtitle":[],"editor":[{"given":"Mehmet","family":"\u00dcnver","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2025,1]]},"references-count":13,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,1]]}},"alternative-id":["10.1155\/jama\/4134128"],"URL":"https:\/\/doi.org\/10.1155\/jama\/4134128","archive":["Portico"],"relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"value":"1110-757X","type":"print"},{"value":"1687-0042","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1]]},"assertion":[{"value":"2024-08-08","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2024-12-30","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-03-10","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"4134128"}}