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 → ∞.
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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).
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).
C. Grebogi, E. Ott, and J. A. Yorke, “Chaotic attractors in crisis,” Phys. Rev. Lett., 48, No. 22, 1507–1510 (1982).
J. Guckenheimer, “One dimensional dynamics,” Ann. New York Acad. Sci., 357, 343–347 (1980).
I. Gumowski and C. Mira, “Accumulations de bifurcations dans une récurrence,” C. R. Acad. Sci. Paris, Sér. A, 281, 45–48 (1975).
I. Gumowski and C. Mira, Dynamique Chaotique. Transformations Ponctuelles. Transition, Ordre-dÉsordre, Cépadués Éditions, Toulouse (1980).
I. Gumowski and C. Mira, “Recurrences and discrete dynamic systems,” Lecture Notes Math., 809, Springer, Berlin (1980).
G. Julia, “Mémoire sur l’itération des fonctions rationnelles,” J. Math. Pures Appl., 4, No. 1, 7ème série, 47–245 (1918).
14. H. Kawakami, “Algorithme optimal définissant les suites de rotation de
H. Kawakami, “Table of rotation sequences of xn+1=xn2-λ”, in: Dynamical Systems and Nonlinear Oscillations (Kyoto, 1985), World Scientific Publ. Co., Singapore (1986), pp. 73–92.
E. N. Lorenz, “Compound windows of the Henon-map,” Phys. D, 237, 1689–1704 (2008).
N. Metropolis, M. L. Stein, and P. R. Stein, “On finite limit sets for transformation on the unit interval,” J. Combin. Theory Ser. A, 15, No. 1, 25–44 (1973).
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.
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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).
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. Mira, “Shrimp fishing, or searching for foliation singularities of the parameter plane. Part I, Basic elements of the parameter plane foliation”, Research Gate Article (2016).
C. Mira, “About intermittency and its different approaches”, Research Gate Article (2019).
C. Mira, “About two approaches of chaotic attractors in crisis”, Research Gate Article (2019).
M. Misiurewicz, “Absolutely continuous measures for certain maps of the interval,” Inst. Hautes Études Sci. Publ. Math., 53, 17–51 (1981).
P. J. Myrberg, “Iteration von Quadratwurzeloperationen”, Ann. Acad. Sci. Fenn., Ser. A. I., 259 (1958).
P. J. Myrberg, “Iteration der reellen Polynome zweiten Grades II”, Ann. Acad. Sci. Fenn., Ser. A. I., 268 (1959).
P. J. Myrberg, “Iteration der reellen Polynome zweiten Grades III”, Ann. Acad. Sci. Fenn., Ser. A. I., 336, 1–10 (1963).
Y. Pomeau and P. Manneville, “Intermittent transition to turbulence in dissipative dynamical systems,” Comm. Math. Phys., 74, 189–197 (1980).
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).
T.-Y. Li and J. A. Yorke, “Period 3 implies chaos,” Amer. Math. Monthly, 82, No. 10, 985–992 (1975).
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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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Mira, C. Fractal Embedded Boxes of Bifurcations. Ukr Math J 76, 80–96 (2024). https://doi.org/10.1007/s11253-024-02309-8
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DOI: https://doi.org/10.1007/s11253-024-02309-8