{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,13]],"date-time":"2026-08-13T13:37:35Z","timestamp":1786628255416,"version":"build-2736575974"},"reference-count":32,"publisher":"AIP Publishing","issue":"8","content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2026,8,1]]},"abstract":"<jats:p>When parameters of a dynamical system change sufficiently fast, critical transitions can take place even in the absence of bifurcations. This phenomenon is known as rate-induced tipping and has been reported in a variety of systems, from simple ordinary differential equations and maps to mathematical models in climate sciences and ecology. In most examples, the transition happens at a critical rate of parameter change, a rate-induced tipping point, and is associated with a simple unstable orbit (edge state). In this work, we explore how this simple picture changes when non-attracting fractal sets exist in the autonomous system, a ubiquitous situation in non-linear dynamics. We show that these fractals in phase-space induce fractals in parameter space, which control the rates and parameter changes that result in tipping. We explain how such rate-induced fractals appear and how the fractal dimensions of the different sets are related to each other. We illustrate our general theory in three paradigmatic systems: a piecewise linear one-dimensional map, the two-dimensional H\u00e9non map, and a forced pendulum.<\/jats:p>","DOI":"10.1063\/5.0324191","type":"journal-article","created":{"date-parts":[[2026,8,13]],"date-time":"2026-08-13T13:03:32Z","timestamp":1786626212000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["Fractals in rate-induced tipping"],"prefix":"10.1063","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-2806-3404","authenticated-orcid":false,"given":"Jason Qianchuan","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, The University of Sydney , Sydney,","place":["Australia"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0006-4230-8785","authenticated-orcid":false,"given":"Yi","family":"Zheng","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, The University of Sydney , Sydney,","place":["Australia"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1932-3710","authenticated-orcid":false,"given":"Eduardo G.","family":"Altmann","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, The University of Sydney , Sydney,","place":["Australia"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2026,8,13]]},"reference":[{"key":"2026081308474562700_c1","doi-asserted-by":"publisher","first-page":"1166","DOI":"10.1098\/rsta.2011.0306","article-title":"Tipping points in open systems: Bifurcation, noise-induced and rate-dependent examples in the climate system","volume":"370","year":"2012","journal-title":"Philos. 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