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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)
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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.

Information

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

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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