Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-12T21:03:14.427Z Has data issue: false hasContentIssue false
  • English
  • Français

Period-doubling cascades galore

Published online by Cambridge University Press:  03 June 2011

EVELYN SANDER and
JAMES A. YORKE
EVELYN SANDER
Affiliation:
Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, VA 22030, USA (email: esander@gmu.edu)
JAMES A. YORKE
Affiliation:
Department of Mathematics, IPST, and Physics Department, University of Maryland, College Park, MD 20742, USA (email: yorke@umd.edu)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

The appearance of numerous period-doubling cascades is among the most prominent features of parametrized maps, that is, smooth one-parameter families of maps F:ℝ×𝔐→𝔐, where 𝔐 is a smooth locally compact manifold without boundary, typically ℝN. Each cascade has infinitely many period-doubling bifurcations, and it is typical to observe that, whenever there are any cascades, there are infinitely many cascades. We develop a general theory of cascades for generic F.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 4 , August 2011, pp. 1249 - 1267
Copyright
Copyright © Cambridge University Press 2011

References

[1]Collet, P. and Eckmann, J.-P.. Iterated Maps on the Interval as Dynamical Systems (Progress in Physics, 1). Birkhäuser, Boston, MA, 1980.Google Scholar
[2]Collet, P., Eckmann, J.-P. and Lanford, III, O. E.. Universal properties of maps on an interval. Comm. Math. Phys. 76(3) (1980), 211–254.CrossRefGoogle Scholar
[3]Collet, P., Eckmann, J.-P. and Koch, H.. On universality for area-preserving maps of the plane. Phys. D 3(3) (1981), 457–467.CrossRefGoogle Scholar
[4]Collet, P., Eckmann, J.-P. and Koch, H.. Period doubling bifurcations for families of maps on R n. J. Stat. Phys. 25(1) (1981), 1–14.Google Scholar
[5]Collet, P., Eckmann, J.-P. and Thomas, L.. A note on the power spectrum of the iterates of Feigenbaum’s function. Comm. Math. Phys. 81(2) (1981), 261–265.CrossRefGoogle Scholar
[6]de Carvalho, A., Lyubich, M. and Martens, M.. Renormalization in the H ′enon family, I: universality but non-rigidity. J. Stat. Phys. 121 (2005), 611–669.Google Scholar
[7]Epstein, H.. New proofs of the existence of the Feigenbaum functions. Comm. Math. Phys. 106(3) (1986), 395–426.Google Scholar
[8]Feigenbaum, M. J.. The universal metric properties of nonlinear transformations. J. Stat. Phys. 21 (1979), 669–706.Google Scholar
[9]Franks, J.. Period doubling and the Lefschetz formula. Trans. Amer. Math. Soc. 287(1) (1985), 275–283.Google Scholar
[10]Gambaudo, J. M., van Strien, S. and Tresser, C.. There exists a C ∞ Kupka–Smale diffeomorphism on S 2 with neither sinks nor sources. Nonlinearity 2 (1989), 287–304.CrossRefGoogle Scholar
[11]Gaidashev, D. and Johnson, T.. Dynamics of the universal area-preserving map associated with period-doubling: stable sets. J. Mod. Dyn. 3 (2009), 555–587.Google Scholar
[12]Gaidashev, D. and Johnson, T.. Dynamics of the universal area-preserving map associated with period doubling: hyperbolic sets. Nonlinearity 22(10) (2009), 2487–2520.CrossRefGoogle Scholar
[13]Kan, I., Kocak, H. and Yorke, J. A.. Antimonotonicity: concurrent creation and annihilation of periodic orbits. Ann. of Math. (2) 136 (1992), 219–252.CrossRefGoogle Scholar
[14]Kuznetsov, S., Mailybaev, A. A. and Sataev, I.. J. Stat. Phys. 130(3) (2008), 599–616.CrossRefGoogle Scholar
[15]Kuznetsov, S. P., Kuznetsov, A. P. and Sataev, I. R.. Multiparameter critical situations, universality and scaling in two-dimensional period-doubling maps. J. Stat. Phys. 121(5–6) (2005), 697–748.CrossRefGoogle Scholar
[16]Lanford, III, O. E.. A computer-assisted proof of the Feigenbaum conjectures. Bull. Amer. Math. Soc. 6 (1982), 427–434.Google Scholar
[17]Mallet-Paret, J. and Yorke, J. A.. Snakes: oriented families of periodic orbits, their sources, sinks, and continuation. J. Differential Equations 43(3) (1982), 419–450.Google Scholar
[18]May, R.. Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos. Science 186 (1974), 645–647.CrossRefGoogle ScholarPubMed
[19]Milnor, J. and Thurston, W.. On iterated maps of the interval. Dynamical Systems (College Park, MD, 1986–87) (Lecture Notes in Mathematics, 1342). Springer, Berlin, 1988, pp. 465–563.Google Scholar
[20]Milnor, J. W.. Topology from the Differentiable Viewpoint, Based on Notes by D. W. Weaver. University Press of Virginia, Charlottesville, VA, 1965.Google Scholar
[21]Myrberg, P.. Sur l’itération des polynomes réels quadratiques. J. Math. Pures Appl. (9) 41 (1962), 339–351.Google Scholar
[22]Palis, J. and Takens, F.. Hyperbolicity and the creation of homoclinic orbits. Ann. of Math. (2) 125 (1987), 337–374.CrossRefGoogle Scholar
[23]Robinson, C.. Dynamical Systems. CRC Press, Boca Raton, FL, 1995.Google Scholar
[24]Joglekar, M. R., Sander, E. and Yorke, J. A.. Fixed points indices and period-doubling cascades. J. Fixed Point Theory Appl. 8 (2010), 151–176.Google Scholar
[25]Yorke, J. A. and Alligood, K. T.. Cascades of period-doubling bifurcations: a prerequisite for horseshoes. Bull. Amer. Math. Soc. (N.S.) 9(3) (1983), 319–322.Google Scholar
[26]Yorke, J. A. and Alligood, K. T.. Period doubling cascades of attractors: a prerequisite for horseshoes. Comm. Math. Phys. 101(3) (1985), 305–321.Google Scholar