{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,3]],"date-time":"2025-09-03T11:09:43Z","timestamp":1756897783800,"version":"3.37.3"},"reference-count":29,"publisher":"Wiley","license":[{"start":{"date-parts":[[2024,2,6]],"date-time":"2024-02-06T00:00:00Z","timestamp":1707177600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002383","name":"King Saud University","doi-asserted-by":"publisher","award":["RSP2024R167"],"award-info":[{"award-number":["RSP2024R167"]}],"id":[{"id":"10.13039\/501100002383","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2024,2,6]]},"abstract":"<jats:p>In this paper, some mathematical properties and dynamic investigations of a Cournot\u2013Bertrand duopoly game using a computed nonlinear cost are studied. The game is repeated and its evolution is presented by noninvertible map. The fixed points for this map are calculated and their stability conditions are discussed. One of those fixed points is Nash equilibrium, and the discussion shows that it can be unstable through flip and Neimark\u2013Sacker bifurcation. The invariant manifold for the game\u2019s map is analyzed. Furthermore, the case when both competing firms are independent is investigated. Due to unsymmetrical structure of the game\u2019s map, global analysis gives rise to complicated basin of attraction for some attracting sets. The topological structure for these basins of attraction shows that escaping (infeasible) domain for some attracting sets becomes unconnected and the rise of holes is obtained. This confirms the existence of contact bifurcation.<\/jats:p>","DOI":"10.1155\/2024\/5594918","type":"journal-article","created":{"date-parts":[[2024,2,6]],"date-time":"2024-02-06T22:20:11Z","timestamp":1707258011000},"page":"1-15","source":"Crossref","is-referenced-by-count":6,"title":["Cournot\u2013Bertrand Duopoly Model: Dynamic Analysis Based on a Computed Cost"],"prefix":"10.1155","volume":"2024","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1167-2430","authenticated-orcid":true,"given":"S. S.","family":"Askar","sequence":"first","affiliation":[{"name":"Department of Statistics and Operations Research College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7103-8872","authenticated-orcid":true,"given":"Ahmed M.","family":"Alshamrani","sequence":"additional","affiliation":[{"name":"Department of Statistics and Operations Research College of Science, King Saud University, P.O. 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