Fractal embedded boxes of bifurcations

UDC 517.9 This descriptive text is essentially based on the Sharkovsky's and Myrberg's publications on the ordering of periodic solutions (cycles) generated by a ${\rm Dim\,}1$ unimodal smooth map $f(x,\lambda).$&nbsp;&nbsp;Taking as an example $f(x,\lambda)=x^{2}-\...

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Дата:	2024
Автор:	Mira, Christian
Формат:	Стаття
Мова:	Англійська
Опубліковано:	Institute of Mathematics, NAS of Ukraine 2024
Онлайн доступ:	https://umj.imath.kiev.ua/index.php/umj/article/view/7661
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Назва журналу:	Ukrains’kyi Matematychnyi Zhurnal
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Ukrains’kyi Matematychnyi Zhurnal

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author	Mira, Christian Mira, Christian
author_facet	Mira, Christian Mira, Christian
author_sort	Mira, Christian
baseUrl_str	https://umj.imath.kiev.ua/index.php/umj/oai
collection	OJS
datestamp_date	2024-06-19T00:35:06Z
description	UDC 517.9 This descriptive text is essentially based on the Sharkovsky's and Myrberg's publications on the ordering of periodic solutions (cycles) generated by a ${\rm Dim\,}1$ unimodal smooth map $f(x,\lambda).$&nbsp;&nbsp;Taking as an example $f(x,\lambda)=x^{2}-\lambda,$&nbsp; it was shown in a paper published in1975 that the bifurcations are organized in the form of a sequence of well-defined fractal embedded ``boxes'' (parameter $\lambda$ intervals), each of which is associated with a basic cycle of period $k$ and a symbol $j$ permitting to distinguish cycles with the same period $k.$&nbsp;Without using the denominations Intermittency (1980) and Attractors in Crisis (1982), this new text shows that the notion of fractal embedded ``boxes'' describes the properties of each of these two situations as the limit of a sequence of well-defined boxes $(k, j)$ as $k\rightarrow\infty.$
doi_str_mv	10.3842/umzh.v76i1.7661
first_indexed	2026-03-24T03:32:59Z
format	Article
fulltext	Skip to main content Advertisement Log in Menu Find a journal Publish with us Track your research Search Saved research Cart Home Ukrainian Mathematical Journal Article Fractal Embedded Boxes of Bifurcations Published: 30 July 2024 Volume 76, pages 80–96, (2024) Cite this article Save article View saved research Ukrainian Mathematical Journal Aims and scope Submit manuscript Christian Mira1 30 Accesses Explore all metrics This descriptive text is essentially based on Sharkovsky’s and Myrberg’s publications on the ordering of periodic solutions (cycles) generated by a Dim 1 unimodal smooth map f(x, λ). Taking f(x, λ) = x2−λ as an example, it was shown in a paper published in 1975 that the bifurcations are organized in the form of a sequence of well-defined fractal embedded “boxes” (parameter λ intervals) each of which is associated with a basic cycle of period k and a symbol j permitting to distinguish cycles with the same period k. Without using the denominations Intermittency (1980) and Attractors in Crisis (1982), this new text shows that the notion of fractal embedded “boxes” describes the properties of each of these two situations as the limit of a sequence of well-defined boxes (k, j) as k → ∞. This is a preview of subscription content, log in via an institution to check access. Access this article Log in via an institution Subscribe and save Springer+ from €37.37 /Month Starting from 10 chapters or articles per month Access and download chapters and articles from more than 300k books and 2,500 journals Cancel anytime View plans Buy Now Buy article PDF 39,95 € Price includes VAT (Ukraine) Instant access to the full article PDF. Institutional subscriptions Similar content being viewed by others Analysis of the non-periodic oscillations of a self-excited friction-damped system with closely spaced modes Article Open access 01 October 2021 Minimal box size for fractal dimension estimation Article Open access 01 June 2016 Test Models for Statistical Inference: Two-Dimensional Reaction Systems Displaying Limit Cycle Bifurcations and Bistability Chapter © 2017 Explore related subjects Discover the latest articles, books and news in related subjects, suggested using machine learning. Differential Equations Dynamical Systems Element cycles Nonlinear Dynamics and Chaos Theory Ordinary Differential Equations Oscillators Dynamical Systems and Ergodic Theory References P. Collet and J. P. Eckmann, Iterated Maps of the Interval as Dynamical Systems, Progress on Physics, Birkhäuser, Boston (1980). J. Couot and C. Mira, “Densités de mesures invariantes non classiques,” C. R. Acad. Sci. Paris, Sér. I, Math., 296, 233–236 (1983). H. El Hamouly and C. Mira, “Singularités dues au feuilletage du plan des bifurcations d’un difféomorphisme bi-dimensionnel,” C. R. Acad. Sci. Paris, Sér. I, Math., 294, 387–390 (1982). H. El Hamouly, Structure des Bifurcations d’un Difféomorphisme Bi-Dimensionnel, Thèse de Docteur-Ingénieur (Math. Appl.), No. 799, Univ. Paul Sabatier, Toulouse (1982). P. Fatou, “Mémoire sur les équations fonctionnelles,” Bull. Soc. Math. France, 47, 161–271 (1919). Article MathSciNet Google Scholar P. Fatou, “Mémoire sur les équations fonctionnelles,” Bull. Soc. Math. France, 48, 33–94 and 208–314 (1920). M. J. Feigenbaum, “Quantitative universality for a class of nonlinear transformations”, J. Stat. Phys., 19, No. 1, 25–52 (1978). Article MathSciNet Google Scholar C. Grebogi, E. Ott, and J. A. Yorke, “Chaotic attractors in crisis,” Phys. Rev. Lett., 48, No. 22, 1507–1510 (1982). Article MathSciNet Google Scholar J. Guckenheimer, “One dimensional dynamics,” Ann. 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Stein, “On finite limit sets for transformation on the unit interval,” J. Combin. Theory Ser. A, 15, No. 1, 25–44 (1973). Article MathSciNet Google Scholar C. Mira, “Accumulations de bifurcations et structures boîtes emboîtées dans les récurrences, ou transformations ponctuelles”, in: Proc. of the VIIth Internat. Conf. on Nonlinear Oscillations (ICNO) (Berlin, Sept. 1975), Akademie-Verlag, Berlin (1977), pp. 81–93. Parser Error (ARS) C. Mira, “Sur la double interprétation, déterministe et statistique, de certaines bifurcations complexes”, C. R. Acad. Sci. Paris, Sér. A, 283, 911–914 (1976). C. Mira, “Frontière floue séparant les domaines d’attraction de deux attracteurs”, C. R. Acad. Sci. Paris, Sér. A, 288, 591–594 (1979). C. Mira, “Sur les points d’accumulation de boîtes appartenant à une strucure boîtes emboitées d’un endomorphisme uni dimensionel”, C. R. Acad. Sci. Paris, Sér. I, Math., 295, 13–16 (1982). C. Mira, Chaotic Dynamics. From the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism, World Scientific Publ. Co., Singapore (1987). Book Google Scholar C. Mira, L. Gardini, A. Barugola, and J. C. Cathala, “Chaotic dynamics in two-dimensional noninvertible maps”, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, 20 (1996). C. Mira, “I. Gumowski and a Toulouse research group in the “prehistoric” times of chaotic dynamics”, Chapter 8 of: “The Chaos Avant-Garde. Memories of the Early Days of Chaos Theory (R. Abraham and Y. Ueda, Eds.), World Sci. Ser. Nonlinear Sci. Ser. A, 39 (2000). C. Mira, “Noninvertible maps”, Publ. on the Website “Scholarpedia”, 2(9), Article 2328 (2007). C. Mira and L. Gardini, “From the box-within-a-box bifurcation structure to the Julia set. II. Bifurcation routes to different Julia sets from an indirect embedding of a quadratic complex map”, Internat. J. Bifur. Chaos Appl. Sci. Eng., 19, No. 10, 3235–3282 (2009). C. 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Phys., 74, 189–197 (1980). Article MathSciNet Google Scholar C. P. Pulkin, “Oscillating iterated sequences”, Doklady Akad. Nauk SSSR (N.S.), 73, No. 6, 1129–1132 (1950). A. N. Sharkovsky, “Coexistence of cycles of a continuous map of a line into itself”, Ukr. Math. Zh., 16, No. 1, 61–71 (1964). Google Scholar T.-Y. Li and J. A. Yorke, “Period 3 implies chaos,” Amer. Math. Monthly, 82, No. 10, 985–992 (1975). Article MathSciNet Google Scholar Download references Author information Authors and Affiliations Groupe d’Etude des Systèmes Non Linéaire et Applications, INSA Toulouse, Toulouse, France Christian Mira Authors Christian MiraView author publications Search author on:PubMed Google Scholar Corresponding author Correspondence to Christian Mira. Additional information Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 76, No. 1, pp. 75–91, January, 2024. Ukrainian DOI: https://doi.org/10.3842/umzh.v76i1.7661. 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spelling	umjimathkievua-article-76612024-06-19T00:35:06Z Fractal embedded boxes of bifurcations Fractal embedded boxes of bifurcations Mira, Christian Mira, Christian EMBEDDED BOXE BIFURCATION UDC 517.9 This descriptive text is essentially based on the Sharkovsky's and Myrberg's publications on the ordering of periodic solutions (cycles) generated by a ${\rm Dim\,}1$ unimodal smooth map $f(x,\lambda).$&nbsp;&nbsp;Taking as an example $f(x,\lambda)=x^{2}-\lambda,$&nbsp; it was shown in a paper published in1975 that the bifurcations are organized in the form of a sequence of well-defined fractal embedded ``boxes'' (parameter $\lambda$ intervals), each of which is associated with a basic cycle of period $k$ and a symbol $j$ permitting to distinguish cycles with the same period $k.$&nbsp;Without using the denominations Intermittency (1980) and Attractors in Crisis (1982), this new text shows that the notion of fractal embedded ``boxes'' describes the properties of each of these two situations as the limit of a sequence of well-defined boxes $(k, j)$ as $k\rightarrow\infty.$ УДК 517.9 Фрактальні вкладені бокси біфуркацій&nbsp; Цей опис в основному базується на публікаціях Шарковського та Мірберга про впорядкування періо\-дичних розв'язків (циклів), що породжені ${\rm Dim\,}1$ унімодальним гладким відображенням&nbsp; $f(x,\lambda).$&nbsp;На прикладі відображення $f(x,\lambda)=x^{2}-\lambda$ у статті 1975 року показано, що біфуркації організовані у вигляді послідовності добре визначених фрактальних вкладених ,,боксів'' (інтервалів параметра $\lambda$), кожен з яких асоціюється з базовим циклом з&nbsp; періодом $k$ і символом $j$, який дозволяє розрізняти цикли з однаковим періодом $k.$&nbsp;Не використовуючи позначень переривчастість (1980) та aтрактори у кризі (1982), цей новий текст показує, що поняття вкладених фрактальних ,,боксів'' описує властивості кожної з цих двох ситуацій як границю послідовності добре визначених боксів $(k, j)$ при $k\rightarrow\infty.$ Institute of Mathematics, NAS of Ukraine 2024-02-02 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7661 10.3842/umzh.v76i1.7661 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 1 (2024); 75 - 91 Український математичний журнал; Том 76 № 1 (2024); 75 - 91 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7661/9680 Copyright (c) 2024 Christian Mira
spellingShingle	Mira, Christian Mira, Christian Fractal embedded boxes of bifurcations
title	Fractal embedded boxes of bifurcations
title_alt	Fractal embedded boxes of bifurcations
title_full	Fractal embedded boxes of bifurcations
title_fullStr	Fractal embedded boxes of bifurcations
title_full_unstemmed	Fractal embedded boxes of bifurcations
title_short	Fractal embedded boxes of bifurcations
title_sort	fractal embedded boxes of bifurcations
topic_facet	EMBEDDED BOXE BIFURCATION
url	https://umj.imath.kiev.ua/index.php/umj/article/view/7661
work_keys_str_mv	AT mirachristian fractalembeddedboxesofbifurcations AT mirachristian fractalembeddedboxesofbifurcations

Fractal embedded boxes of bifurcations

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