Bifurcation to badly ordered orbits in one-parameter families of circle maps, or angels fallen from the devil’s staircase

Hockett, Kevin; Holmes, Philip

doi:10.1090/S0002-9939-1988-0934888-7

Bifurcation to badly ordered orbits in one-parameter families of circle maps, or angels fallen from the devil’s staircase
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by Kevin Hockett and Philip Holmes

Proc. Amer. Math. Soc. 102 (1988), 1031-1051

DOI: https://doi.org/10.1090/S0002-9939-1988-0934888-7

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Abstract:

We discuss the structure of the bifurcation set of a one-parameter family of endomorphisms of ${S^1}$ having two critical points and negative Schwarzian derivative. We concentrate on the case in which one of the endpoints of the rotation set is rational, providing a partial characterization of components of the nonwandering set having specified rotation number and the bifurcations in which they are created. In particular we find, for each rational rotation number $p’/q’$ less than the upper boundary of the rotation set $p/q$, infinitely many saddle-node bifurcations to badly ordered periodic orbits of rotation number $p’/q’$.

References

Serge Aubry, The twist map, the extended Frenkel-Kontorova model and the devil’s staircase, Phys. D 7 (1983), no. 1-3, 240–258. Order in chaos (Los Alamos, N.M., 1982). MR 719055, DOI 10.1016/0167-2789(83)90129-X
S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions. I. Exact results for the ground-states, Phys. D 8 (1983), no. 3, 381–422. MR 719634, DOI 10.1016/0167-2789(83)90233-6
C. Bernhardt, Rotation intervals of a class of endomorphisms of the circle, Proc. London Math. Soc. (3) 45 (1982), no. 2, 258–280. MR 670037, DOI 10.1112/plms/s3-45.2.258
George D. Birkhoff, Sur quelques courbes fermées remarquables, Bull. Soc. Math. France 60 (1932), 1–26 (French). MR 1504983, DOI 10.24033/bsmf.1182
Louis Block and John Franke, Existence of periodic points for maps of $S^{1}$, Invent. Math. 22 (1973/74), 69–73. MR 358867, DOI 10.1007/BF01425575

Bifurcations of endomorphisms of

Louis Block, Periodic orbits of continuous mappings of the circle, Trans. Amer. Math. Soc. 260 (1980), no. 2, 553–562. MR 574798, DOI 10.1090/S0002-9947-1980-0574798-8
Louis Block, John Guckenheimer, MichałMisiurewicz, and Lai Sang Young, Periodic points and topological entropy of one-dimensional maps, Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979) Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 18–34. MR 591173
Louis Block, Periods of periodic points of maps of the circle which have a fixed point, Proc. Amer. Math. Soc. 82 (1981), no. 3, 481–486. MR 612745, DOI 10.1090/S0002-9939-1981-0612745-7
Philip L. Boyland and Glen Richard Hall, Invariant circles and the order structure of periodic orbits in monotone twist maps, Topology 26 (1987), no. 1, 21–35. MR 880504, DOI 10.1016/0040-9383(87)90017-6
Philip L. Boyland, Bifurcations of circle maps: Arnol′d tongues, bistability and rotation intervals, Comm. Math. Phys. 106 (1986), no. 3, 353–381. MR 859816, DOI 10.1007/BF01207252

Periodic orbits for Birkhoff attractors

Alain Chenciner, Jean-Marc Gambaudo, and Charles Tresser, Une remarque sur la structure des endomorphismes de degré $1$ du cercle, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 5, 145–148 (French, with English summary). MR 756312
Alain Chenciner, Jean-Marc Gambaudo, and Charles Tresser, Une remarque sur les familles d’endomorphismes de degré $1$ du cercle, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 15, 771–773. MR 772091
Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338
Pierre Collet and Jean-Pierre Eckmann, Iterated maps on the interval as dynamical systems, Modern Birkhäuser Classics, Birkhäuser Boston, Ltd., Boston, MA, 2009. Reprint of the 1980 edition. MR 2541754, DOI 10.1007/978-0-8176-4927-2
Robert L. Devaney, An introduction to chaotic dynamical systems, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1986. MR 811850
Jean-Marc Gambaudo, Oscar Lanford III, and Charles Tresser, Dynamique symbolique des rotations, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 16, 823–826 (French, with English summary). MR 772104
John Guckenheimer, On the bifurcation of maps of the interval, Invent. Math. 39 (1977), no. 2, 165–178. MR 438399, DOI 10.1007/BF01390107
John Guckenheimer, Sensitive dependence to initial conditions for one-dimensional maps, Comm. Math. Phys. 70 (1979), no. 2, 133–160. MR 553966, DOI 10.1007/BF01982351
John Guckenheimer, Bifurcations of dynamical systems, Dynamical systems (C.I.M.E. Summer School, Bressanone, 1978) Progr. Math., vol. 8, Birkhäuser, Boston, Mass., 1980, pp. 115–231. MR 589591
John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. MR 709768, DOI 10.1007/978-1-4612-1140-2

Recurrences and discrete dynamical systems

Bifurcations of rational and irrational Cantor sets and periodic orbits in maps of the circle

Josephson’s junction, annulus maps, Birkhoff attractors, horseshoes and rotation sets

6

Bifurcations to rotating Cantor sets in maps of the circle

P. Holmes and D. Whitley, Bifurcations of one- and two-dimensional maps, Philos. Trans. Roy. Soc. London Ser. A 311 (1984), no. 1515, 43–102. MR 776547, DOI 10.1098/rsta.1984.0020
Ryuichi Ito, Rotation sets are closed, Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 1, 107–111. MR 591976, DOI 10.1017/S0305004100057984

On smooth mappings of the interval into itself

14

Leo Jonker and David Rand, Bifurcations in one dimension. I. The nonwandering set, Invent. Math. 62 (1981), no. 3, 347–365. MR 604832, DOI 10.1007/BF01394248
Leo Jonker and David Rand, Bifurcations in one dimension. I. The nonwandering set, Invent. Math. 62 (1981), no. 3, 347–365. MR 604832, DOI 10.1007/BF01394248
Leo P. Kadanoff, Supercritical behavior of an ordered trajectory, J. Statist. Phys. 31 (1983), no. 1, 1–27. MR 711468, DOI 10.1007/BF01010921
A. Katok, Some remarks of Birkhoff and Mather twist map theorems, Ergodic Theory Dynam. Systems 2 (1982), no. 2, 185–194 (1983). MR 693974, DOI 10.1017/s0143385700001504
A. Katok, Periodic and quasiperiodic orbits for twist maps, Dynamical systems and chaos (Sitges/Barcelona, 1982) Lecture Notes in Phys., vol. 179, Springer, Berlin, 1983, pp. 47–65. MR 708545, DOI 10.1007/3-540-12276-1_{3}

Existence d’orbites quasi-periodiques dansles attracteurs de Birkhoff

Transition to chaos for two-frequency systems

Boundary of chaos for bimodal maps of the interval

R. S. MacKay and C. Tresser, Transition to topological chaos for circle maps, Phys. D 19 (1986), no. 2, 206–237. MR 844701, DOI 10.1016/0167-2789(86)90020-5
John N. Mather, Existence of quasiperiodic orbits for twist homeomorphisms of the annulus, Topology 21 (1982), no. 4, 457–467. MR 670747, DOI 10.1016/0040-9383(82)90023-4

Non-uniqueness of solutions of Percival’s Euler-Lagrange equation

86

John N. Mather, More Denjoy minimal sets for area preserving diffeomorphisms, Comment. Math. Helv. 60 (1985), no. 4, 508–557. MR 826870, DOI 10.1007/BF02567431
Robion C. Kirby and Laurence C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Annals of Mathematics Studies, No. 88, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977. With notes by John Milnor and Michael Atiyah. MR 0645390, DOI 10.1515/9781400881505
MichałMisiurewicz, Structure of mappings of an interval with zero entropy, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 5–16. MR 623532, DOI 10.1007/BF02698685
MichałMisiurewicz, Absolutely continuous measures for certain maps of an interval, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 17–51. MR 623533, DOI 10.1007/BF02698686
MichałMisiurewicz, Twist sets for maps of the circle, Ergodic Theory Dynam. Systems 4 (1984), no. 3, 391–404. MR 776876, DOI 10.1017/S0143385700002534
S. Newhouse, J. Palis, and F. Takens, Bifurcations and stability of families of diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 57 (1983), 5–71. MR 699057, DOI 10.1007/BF02698773
Zbigniew Nitecki, Topological dynamics on the interval, Ergodic theory and dynamical systems, II (College Park, Md., 1979/1980), Progr. Math., vol. 21, Birkhäuser, Boston, Mass., 1982, pp. 1–73. MR 670074
Sebastian J. van Strien, On the bifurcations creating horseshoes, Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980), Lecture Notes in Math., vol. 898, Springer, Berlin-New York, 1981, pp. 316–351. MR 654898
David Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math. 35 (1978), no. 2, 260–267. MR 494306, DOI 10.1137/0135020