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These features are usually investigated quantitatively by studying the properties of the underlying attractor, the compact object asymptotically hosting the trajectories of the system with their invariant density in the phase space. This multi-scale nature of natural systems makes it practically impossible to get a clear picture of the attracting set. Indeed, it spans over a wide range of spatial scales and may even change in time due to non-stationary forcing. Here, we combine an adaptive decomposition method with extreme value theory to study the properties of the instantaneous scale-dependent dimension, which has been recently introduced to characterize such temporal and spatial scale-dependent attractors in turbulence and astrophysics. To provide a quantitative analysis of the properties of this metric, we test it on the well-known low-dimensional deterministic Lorenz-63 system perturbed with additive or multiplicative noise. We demonstrate that the properties of the invariant set depend on the scale we are focusing on and that the scale-dependent dimensions can discriminate between additive and multiplicative noise despite the fact that the two cases have exactly the same stationary invariant measure at large scales. The proposed formalism can be generally helpful to investigate the role of multi-scale fluctuations within complex systems, allowing us to deal with the problem of characterizing the role of stochastic fluctuations across a wide range of physical systems.<\/jats:p>","DOI":"10.1063\/5.0106053","type":"journal-article","created":{"date-parts":[[2023,2,27]],"date-time":"2023-02-27T17:56:50Z","timestamp":1677520610000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":15,"title":["Scale dependence of fractal dimension in deterministic and stochastic Lorenz-63 systems"],"prefix":"10.1063","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6096-0220","authenticated-orcid":false,"given":"T.","family":"Alberti","sequence":"first","affiliation":[{"name":"INAF-Istituto di Astrofisica e Planetologia Spaziali 1 , via del Fosso del Cavaliere 100, 00133 Roma, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5001-5698","authenticated-orcid":false,"given":"D.","family":"Faranda","sequence":"additional","affiliation":[{"name":"Laboratoire des Sciences du Climat et de l\u2019Environnement, CEA Saclay l\u2019Orme des Merisiers, UMR 8212 CEA-CNRS-UVSQ, Universit\u00e9 Paris-Saclay, and IPSL 2 , 91191 Gif-sur-Yvette, France"},{"name":"London Mathematical Laboratory 3 , 8 Margravine Gardens, London W6 8RH, United Kingdom"},{"name":"LMD\/IPSL, Ecole Normale Superieure, PSL Research University 4 , 75005 Paris, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9392-1471","authenticated-orcid":false,"given":"V.","family":"Lucarini","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, University of Reading 5 , RG6 6AH Reading, United Kingdom"},{"name":"Centre for the Mathematics of Planet Earth, University of Reading 6 , RG6 6AX Reading, United Kingdom"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7023-6375","authenticated-orcid":false,"given":"R. 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