{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,12]],"date-time":"2026-01-12T23:30:11Z","timestamp":1768260611688,"version":"3.49.0"},"reference-count":39,"publisher":"World Scientific Pub Co Pte Ltd","issue":"12","funder":[{"DOI":"10.13039\/501100008227","name":"University of South Africa","doi-asserted-by":"crossref","award":["None"],"award-info":[{"award-number":["None"]}],"id":[{"id":"10.13039\/501100008227","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2020,9,30]]},"abstract":"<jats:p> It is not common in applied sciences to realize simulations which depict fractal representation in attractors\u2019 dynamics, the reason being a combination of many factors including the nature of the phenomenon that is described and the type of differential operator used in the system. In this work, we use the fractal-fractional derivative with a fractional order to analyze the modified proto-Lorenz system that is usually characterized by chaotic attractors with many scrolls. The fractal-fractional operator used in this paper is a combination of fractal process and fractional differentiation, which is a relatively new concept with most of the properties and features still to be known. We start by summarizing the basic notions related to the fractal-fractional operator. After that, we enumerate the main points related to the establishment of proto-Lorenz system\u2019s equations, leading to the [Formula: see text]th cover of the proto-Lorenz system that contains [Formula: see text] scrolls ([Formula: see text]). The triple and quadric cover of the resulting fractal and fractional proto-Lorenz system are solved using the Haar wavelet methods and numerical simulations are performed. Due to the impact of the fractal-fractional operator, the system is able to maintain its chaotic state of attractor with many scrolls. Additionally, such attractor can self-replicate in a fractal process as the derivative order changes. This result reveals another great feature of the fractal-fractional derivative with fractional order. <\/jats:p>","DOI":"10.1142\/s0218127420501801","type":"journal-article","created":{"date-parts":[[2020,10,5]],"date-time":"2020-10-05T08:54:32Z","timestamp":1601888072000},"page":"2050180","source":"Crossref","is-referenced-by-count":20,"title":["The Proto-Lorenz System in Its Chaotic Fractional and Fractal Structure"],"prefix":"10.1142","volume":"30","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6520-1039","authenticated-orcid":false,"given":"Emile Franc","family":"Doungmo Goufo","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, University of South Africa, Florida Campus, Florida, 0003, South Africa"}]}],"member":"219","published-online":{"date-parts":[[2020,10,3]]},"reference":[{"key":"S0218127420501801BIB001","doi-asserted-by":"publisher","DOI":"10.2298\/TSCI160111018A"},{"key":"S0218127420501801BIB002","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2017.04.027"},{"key":"S0218127420501801BIB003","doi-asserted-by":"publisher","DOI":"10.1140\/epjp\/i2019-12777-8"},{"key":"S0218127420501801BIB004","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2008.07.003"},{"key":"S0218127420501801BIB005","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.98.178301"},{"key":"S0218127420501801BIB007","first-page":"1","volume":"1","author":"Caputo M.","year":"2015","journal-title":"Progr. 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