Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-26T19:05:15.544Z Has data issue: false hasContentIssue false
  • English
  • Français

A classification of explosions in dimension one

Published online by Cambridge University Press:  01 April 2009

E. SANDER  and
J. A. YORKE
E. SANDER
Affiliation:
Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, VA 22030, USA (email: sander@math.gmu.edu)
J. A. YORKE
Affiliation:
IPST, University of Maryland, College Park, MD 20742, USA (email: yorke@ipst.umd.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

A discontinuous change in the size of an attractor is the most easily observed type of global bifurcation. More generally, an explosion is a discontinuous change in the set of recurrent points. An explosion often results from heteroclinic and homoclinic tangency bifurcations. We prove that, for one-dimensional maps, explosions are generically the result of either tangency or saddle-node bifurcations. Furthermore, we give necessary and sufficient conditions for generic tangency bifurcations to lead to explosions.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 2 , April 2009 , pp. 715 - 731
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Alligood, K., Sander, E. and Yorke, J.. Explosions: global bifurcations at heteroclinic tangencies. Ergod. Th. & Dynam. Sys. 22(4) (2002), 953–972.CrossRefGoogle Scholar
[2]Alligood, K., Sander, E. and Yorke, J.. Three-dimensional crisis: crossing bifurcations and unstable dimension variability. Phys. Rev. Lett. 96(244103) (2006).CrossRefGoogle ScholarPubMed
[3]Alsedà, L., López, V. J. and Snoha, L.. All solenoids of piecewise smooth maps are period doubling. Fund. Math. 157(2–3) (1998), 121–138. (Dedicated to the memory of Wiesław Szlenk.)CrossRefGoogle Scholar
[4]Block, L.. Homoclinic points of mappings of the interval. Proc. Amer. Math. Soc. 72(3) (1978), 576–580.CrossRefGoogle Scholar
[5]Block, L. and Coppel, W.. Dynamics in One Dimension (Lecture Notes in Mathematics, 1513). Springer, Berlin, 1992.CrossRefGoogle Scholar
[6]Block, L. and Hart, D.. The bifurcation of homoclinic orbits of maps of the interval. Ergod. Th. & Dynam. Sys. 2(2) (1982), 131–138.CrossRefGoogle Scholar
[7]Blokh, A.. Density of periodic orbits in ω-limit sets with the Hausdorff metric. Real Anal. Exchange 24(2) (1998/99), 503–521.CrossRefGoogle Scholar
[8]Blokh, A., Bruckner, A., Humke, P. and Smítal, J.. The space of ω-limit sets of a continuous map of the interval. Trans. Amer. Math. Soc. 348(4) (1996), 1357–1372.CrossRefGoogle Scholar
[9]Blokh, A. M.. The ‘spectral’ decomposition for one-dimensional maps. Dynamics Reported (Dynamics Reported, Expositions in Dynamical Systems (New Series), 4). Springer, Berlin, 1995, pp. 1–59.Google Scholar
[10]Bonatti, C., Diaz, L. and Viana, M.. Dynamics Beyond Hyperbolicity. Springer, Berlin, 2005.Google Scholar
[11]Bowen, R.. Markov partitions for axiom a diffeomorphisms. Amer. J. Math. 92 (1970), 725–747.CrossRefGoogle Scholar
[12]Conley, C.. Isolated Invariant Sets and the Morse Index. American Mathematical Society, Providence, RI, 1978.CrossRefGoogle Scholar
[13]Díaz, L. and Rocha, J.. Heterodimensional cycles, partial hyperbolicity and limit dynamics. Fund. Math. 174(2) (2002), 127–186.CrossRefGoogle Scholar
[14]Horita, V., Muniz, N. and Sabini, P. R.. Non-periodic bifurcations of one-dimensional maps. Ergod. Th. & Dynam. Sys. 27(2) (2007), 459–492.CrossRefGoogle Scholar
[15]López, V. J.. Period doubling is the boundary of chaos and of order in the C 1-topology of interval maps. Nonlinearity 15(3) (2002), 817–839.CrossRefGoogle Scholar
[16]Mañé, R.. Hyperbolicity, sinks and measure in one-dimensional dynamics. Comm. Math. Phys. 100(4) (1985), 495–524.CrossRefGoogle Scholar
[17]Marotto, F.. Snap-back repellers imply chaos in R n. J. Math. Anal. Appl. 63 (1978), 199–223.CrossRefGoogle Scholar
[18]Martens, M., de Melo, W. and van Strien, S.. Julia-Fatou-Sullivan theory for real one-dimensional dynamics. Acta Math. 168(3–4) (1992), 273–318.CrossRefGoogle Scholar
[19]Newhouse, S. and Palis, J.. Cycles and bifurcation theory. Astérisque 31(44–140) (1976).Google Scholar
[20]Palis, J. and Takens, F.. Hyperbolicity and the creation of homoclinic orbits. Ann. of Math. (2) 125 (1987), 337–374.CrossRefGoogle Scholar
[21]Palis, J. and Takens, F.. Hyperbolicity & Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Cambridge University Press, Cambridge, 1993.Google Scholar
[22]Patterson, S. E.. Ω-stable limit set explosions. Trans. Amer. Math. Soc. 294(2) (1986), 775–798.Google Scholar
[23]Robert, C., Alligood, K. T., Ott, E. and Yorke, J. A.. Explosions of chaotic sets. Phys. D 144(1–2) (2000), 44–61.CrossRefGoogle Scholar
[24]Sabini, P. R.. Non-periodic bifurcations at the boundary of hyperbolic systems. PhD Thesis, IMPA, 2001.Google Scholar
[25]Sander, E.. Homoclinic tangles for noninvertible maps. Nonlinear Anal. 41(1–2) (2000), 259–276.CrossRefGoogle Scholar