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The results show that, despite its relatively simple architecture of neural network, the trained map is capable of capturing the key dynamical features of the original oscillators, even though corresponding examples are absent from the training data. In a two-dimensional parameter space, the neural-network map reproduces, without direct exposure during training, the multistability of the oscillators and their bifurcations, including the Andronov\u2013Hopf bifurcation, the saddle-node bifurcation associated with the birth of a limit cycle, and the codimension-2 Bautin bifurcation. We further show that, although explicit examples of these effects are not necessary, the ability to recover them still depends on an appropriate choice of training data.<\/jats:p>","DOI":"10.1063\/5.0332036","type":"journal-article","created":{"date-parts":[[2026,5,15]],"date-time":"2026-05-15T12:05:37Z","timestamp":1778846737000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["Neural-network maps for two-parameter modeling of bistability and codimension-two bifurcations in two-dimensional flow dynamical systems"],"prefix":"10.1063","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2685-9828","authenticated-orcid":false,"given":"Pavel V.","family":"Kuptsov","sequence":"first","affiliation":[{"name":"Laboratory of Topological Methods in Dynamics, HSE University , 25\/12 Bolshaya Pecherskaya str., Nizhny Novgorod 603155,","place":["Russian Federation"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-0614-073X","authenticated-orcid":false,"given":"Anton A.","family":"Panyushev","sequence":"additional","affiliation":[{"name":"Laboratory of Topological Methods in Dynamics, HSE University , 25\/12 Bolshaya Pecherskaya str., Nizhny Novgorod 603155,","place":["Russian Federation"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4781-0567","authenticated-orcid":false,"given":"Nataliya V.","family":"Stankevich","sequence":"additional","affiliation":[{"name":"Laboratory of Topological Methods in Dynamics, HSE University , 25\/12 Bolshaya Pecherskaya str., Nizhny Novgorod 603155,","place":["Russian Federation"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"317","published-online":{"date-parts":[[2026,5,15]]},"reference":[{"key":"2026051508053295200_c1","volume-title":"Deterministic Chaos: An Introduction","year":"2006","edition":"4th"},{"key":"2026051508053295200_c2","volume-title":"Chaotic Synchronization: Applications to Living Systems","year":"2002"},{"key":"2026051508053295200_c3","volume-title":"Chaos in Nature","year":"2019","edition":"2nd"},{"issue":"2","key":"2026051508053295200_c4","doi-asserted-by":"publisher","first-page":"291","DOI":"10.1134\/S1560354725020078","article-title":"Verification of chaos in a human cardiovascular system model","volume":"30","year":"2025","journal-title":"Regul. 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