{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,19]],"date-time":"2025-11-19T07:00:20Z","timestamp":1763535620936,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2019,4,21]],"date-time":"2019-04-21T00:00:00Z","timestamp":1555804800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61703251"],"award-info":[{"award-number":["61703251"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2017M612337"],"award-info":[{"award-number":["2017M612337"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Scientific and Technological Planning Projects of Universities in Shandong Province","award":["J18KB097"],"award-info":[{"award-number":["J18KB097"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper investigates the destruction of the symmetrical structure of the noise-perturbed Mandelbrot set (M-set). By applying the \u201csymmetry criterion\u201d method, we quantitatively compare the damages to the symmetry of the noise-perturbed Mandelbrot set resulting from additive and multiplicative noises. Because of the uneven distribution between the core positions and the edge positions of the noise-perturbed Mandelbrot set, the comparison results reveal a paradox between the visual sense and quantified result. Thus, we propose a new \u201cvisual symmetry criterion\u201d method that is more suitable for the measurement of visual asymmetry.<\/jats:p>","DOI":"10.3390\/sym11040577","type":"journal-article","created":{"date-parts":[[2019,4,22]],"date-time":"2019-04-22T11:02:53Z","timestamp":1555930973000},"page":"577","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["The Symmetry in the Noise-Perturbed Mandelbrot Set"],"prefix":"10.3390","volume":"11","author":[{"given":"Tianwen","family":"Sun","sequence":"first","affiliation":[{"name":"Research Center of Dynamics System and Control Science, Shandong Normal University, Jinan 250014, China"}]},{"given":"Da","family":"Wang","sequence":"additional","affiliation":[{"name":"Research Center of Dynamics System and Control Science, Shandong Normal University, Jinan 250014, China"}]}],"member":"1968","published-online":{"date-parts":[[2019,4,21]]},"reference":[{"key":"ref_1","first-page":"47","article-title":"M\u00e8moire sur l\u2019it\u00e8ration des fonctions rationnelles","volume":"8","author":"Julia","year":"1918","journal-title":"J. Math. Pures Appl."},{"key":"ref_2","unstructured":"Mandelbrot, B.B. (2013). Fractals and Chaos: The Mandelbrot Set and Beyond, Springer Science & Business Media."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1007\/BF02098448","article-title":"Similarity between the Mandelbrot set and Julia sets","volume":"134","author":"Lei","year":"1990","journal-title":"Commun. Math. Phys."},{"key":"ref_4","first-page":"693","article-title":"Fractal structures of the non-boundary region of the generalized Mandelbrot set","volume":"11","author":"Wang","year":"2001","journal-title":"Prog. Nat. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1148","DOI":"10.22436\/jnsa.010.03.24","article-title":"Fractal generation method based on asymptote family of generalized Mandelbrot set and its application","volume":"10","author":"Liu","year":"2017","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"843","DOI":"10.1007\/s11071-013-0836-5","article-title":"Hyperdimensional generalized M-J sets in hypercomplex number space","volume":"73","author":"Wang","year":"2013","journal-title":"Nonlinear Dyn."},{"key":"ref_7","unstructured":"Peitgen, H.O., and Saupe, D. (2011). The Science of Fractal Images, Springer Publishing Company, Incorporated."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1007\/s11071-010-9845-9","article-title":"Synchronization and coupling of Mandelbrot sets","volume":"64","author":"Zhang","year":"2011","journal-title":"Nonlinear Dyn."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1795","DOI":"10.1007\/s11071-016-2606-7","article-title":"Control of the spatial Mandelbrot set generated in coupled map lattice","volume":"84","author":"Wang","year":"2016","journal-title":"Nonlinear Dyn."},{"key":"ref_10","first-page":"279","article-title":"Superior Mandelbrot set","volume":"8","author":"Rani","year":"2004","journal-title":"J. Korea Soc. Math. Educ. Ser. D Res. Math. Educ."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1142\/S0218348X0000041X","article-title":"A generalized Mandelbrot set for bicomplex numbers","volume":"8","author":"Rochon","year":"2010","journal-title":"Fractals"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/S0167-2789(98)00243-7","article-title":"Physical meaning for Mandelbrot and Julia sets","volume":"125","author":"Beck","year":"1999","journal-title":"Phys. Nonlinear Phenom."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/j.biosystems.2012.04.005","article-title":"Morphogenetic fields in embryogenesis, regeneration, and cancer: Non-local control of complex patterning","volume":"109","author":"Levin","year":"2012","journal-title":"Biosystems"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1080\/00207160.2015.1130825","article-title":"Fractal analysis and control in the predator-prey model","volume":"94","author":"Sun","year":"2017","journal-title":"Int. J. Comput. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1049\/iet-ipr.2014.0224","article-title":"Image compression and encryption scheme using fractal dictionary and Julia set","volume":"9","author":"Sun","year":"2014","journal-title":"Image Process. IET"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"955","DOI":"10.1016\/j.physa.2018.06.100","article-title":"Discrete chaotic maps obtained by symmetric integration","volume":"509","author":"Butusov","year":"2018","journal-title":"Phys. Stat. Mech. Its Appl."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Butusov, D.N., Karimov, A.I., and Tutueva, A.V. (2016, January 2\u20133). Symmetric extrapolation solvers for ordinary differential equations. Proceedings of the 2016 IEEE NW Russia Young Researchers in Electrical and Electronic Engineering Conference, St. Petersburg, Russia.","DOI":"10.1109\/EIConRusNW.2016.7448145"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1493","DOI":"10.1140\/epjst\/e2015-02475-x","article-title":"Multistability in symmetric chaotic systems","volume":"224","author":"Li","year":"2015","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1088\/0951-7715\/15\/4\/309","article-title":"On the Mandelbrot set for pairs of linear maps","volume":"15","author":"Bandt","year":"2002","journal-title":"Nonlinearity"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Wang, D., and Liu, S.T. (2016). On the boundedness and symmetry properties of the fractal sets generated from alternated complex map. Symmetry, 8.","DOI":"10.3390\/sym8020007"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2067","DOI":"10.1016\/S0960-0779(99)00101-0","article-title":"On the Julia set of the perturbed Mandelbrot map","volume":"11","author":"Argyris","year":"2000","journal-title":"Chaos Solitons Fractals"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/S0960-0779(00)00257-5","article-title":"On the Julia sets of a noise-perturbed Mandelbrot map","volume":"13","author":"Argyris","year":"2002","journal-title":"Chaos Solitons Fractals"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1131","DOI":"10.1016\/S0960-0779(99)00017-X","article-title":"On perturbations of the Mandelbrot map","volume":"11","author":"Argyris","year":"2000","journal-title":"Chaos Solitons Fractals"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/j.jmaa.2008.04.032","article-title":"Noise perturbed generalized Mandelbrot sets","volume":"347","author":"Wang","year":"2008","journal-title":"J. Math. Anal. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/j.amc.2008.12.017","article-title":"The generalized Mandelbrot set perturbed by composing noise of additive and multiplicative","volume":"210","author":"Wang","year":"2009","journal-title":"Appl. Math. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/s10851-009-0171-0","article-title":"Effect of stochastic noise on superior Julia sets","volume":"36","author":"Rani","year":"2010","journal-title":"J. Math. Imaging Vis."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1883","DOI":"10.1007\/s11071-011-0115-2","article-title":"Dynamic noise perturbed generalized superior Mandelbrot sets","volume":"67","author":"Agarwal","year":"2012","journal-title":"Nonlinear Dyn."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3674","DOI":"10.1016\/j.amc.2009.11.006","article-title":"On a topological closeness of perturbed Mandelbrot sets","volume":"215","author":"Andreadis","year":"2010","journal-title":"Appl. Math. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1250221","DOI":"10.1142\/S0218127412502215","article-title":"On a closeness of the Julia sets of noise-perturbed complex quadratic maps","volume":"22","author":"Andreadis","year":"2012","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1016\/j.cnsns.2017.03.001","article-title":"On the noise-perturbed spatial Julia set generated by Lorenz system","volume":"50","author":"Wang","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1295","DOI":"10.1007\/s11071-016-3115-4","article-title":"Real and complex behavior for networks of coupled logistic maps","volume":"87","author":"Pignatelli","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"R\u0103dulescu, A., and Evans, S. (2018). Asymptotic sets in networks of coupled quadratic nodes. J. Complex Netw.","DOI":"10.1093\/comnet\/cny021"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/4\/577\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:46:09Z","timestamp":1760186769000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/4\/577"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4,21]]},"references-count":32,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2019,4]]}},"alternative-id":["sym11040577"],"URL":"https:\/\/doi.org\/10.3390\/sym11040577","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,4,21]]}}}