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Periodic points and chaotic functions in the unit interval

Published online by Cambridge University Press:  17 April 2009

G.J. Butler  and
G. Pianigiani
G.J. Butler
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada;
G. Pianigiani
Affiliation:
Istituto Matematico “Ulisse Dini”, University di Firenze, Firenze, Italy.
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Abstract

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It is shown that the set of chaotic self-maps of the unit interval contains an open dense subset of the space of all continuous self-maps of the unit interval. Other aspects of chaotic behaviour are also considered together with some illustrative examples.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 18 , Issue 2 , April 1978 , pp. 255 - 265
Copyright
Copyright © Australian Mathematical Society 1978

References

[1]Guckenheimer, John, “On the bifurcation of maps of the interval”, Invent. Math. 39 (1977), 165–178.CrossRefGoogle Scholar
[2]Guckenheimer, J., Oster, G. and Ipaktchi, A., “The dynamics of density-dependent population models”, J. Math. Biology 4 (1977), 101–147.CrossRefGoogle ScholarPubMed
[3]Hoppensteadt, F.C. and Hyman, J.M., “Periodic solutions of a logistic difference equation”, SIAM J. Appl. Math. 32 (1977), 73–81.CrossRefGoogle Scholar
[4]Kloeden, Peter E., “Chaotic difference equations are dense”, Bull. Austral. Math. Soc. 15 (1976), 371–379.CrossRefGoogle Scholar
[5]Lasota, A., Pianigiani, G., “Invariant measures on topological spaces”, Boll. Un. Mat. Ital. (to appear).Google Scholar
[6]Li, Tien-Yien and Yorke, James A., “Period three implies chaos”, Amer. Math. Monthly 82 (1975), 985–992.CrossRefGoogle Scholar
[7]May, Robert M., “Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos”, Science 186 (1974), 645–647.CrossRefGoogle ScholarPubMed
[8]Ruelle, David and Takens, Floris, “On the nature of turbulence”, Comm. Math. Phys. 20 (1971), 167–192.CrossRefGoogle Scholar
[9]шapHoβCHий, A.H. [Sharkovsky, A.N.], “CocyщeCTβOβaHциHлO⊓pepbiβHoΓo ⊓peoбpaзOβaHиe ⊓peMOй β ceб” [Co-existence of the cycles of a continuous mapping of the line into itself], Ukrain. Mat. ž. 16 (1964), 61–71.Google Scholar
[10]Smale, S., and Williams, R.F., “The qualitative analysis of a difference equation of population growth”, J. Math. Biology 3 (1976), 1–4.CrossRefGoogle ScholarPubMed