{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:01:20Z","timestamp":1760234480584,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,5,25]],"date-time":"2021-05-25T00:00:00Z","timestamp":1621900800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We use the natural invariant density of the map and the Perron\u2013Frobenius operator to analytically evaluate the statistical properties for chaotic intermittency. This study can be understood as an improvement of the previous ones because it does not introduce assumptions about the reinjection probability density function in the laminar interval or the map density at pre-reinjection points. To validate the new theoretical equations, we study a symmetric map and a non-symmetric one. The cusp map has symmetry about x=0, but the Manneville map has no symmetry. We carry out several comparisons between the theoretical equations here presented, the M function methodology, the classical theory of intermittency, and numerical data. The new theoretical equations show more accuracy than those calculated with other techniques.<\/jats:p>","DOI":"10.3390\/sym13060935","type":"journal-article","created":{"date-parts":[[2021,5,25]],"date-time":"2021-05-25T22:02:23Z","timestamp":1621980143000},"page":"935","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Calculation of the Statistical Properties in Intermittency Using the Natural Invariant Density"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7250-0392","authenticated-orcid":false,"given":"Sergio","family":"Elaskar","sequence":"first","affiliation":[{"name":"Departamento de Aeron\u00e1utica, Instituto de Estudios Avanzados en Ingenier\u00eda y Tecnolog\u00eda (IDIT), FCEFyN, Universidad Nacional de C\u00f3rdoba and CONICET, C\u00f3rdoba 5000, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ezequiel","family":"del R\u00edo","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica Aplicada, ETSIAE, Universidad Polit\u00e9cnica de Madrid, 28040 Madrid, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8565-5193","authenticated-orcid":false,"given":"Denis","family":"Lorenz\u00f3n","sequence":"additional","affiliation":[{"name":"Instituto de Estudios Avanzados en Ingenier\u00eda y Tecnolog\u00eda (IDIT), Universidad Nacional de C\u00f3rdoba and CONICET, C\u00f3rdoba 5000, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Schuster, H., and Just, W. 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