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Chaotic difference equations are dense

Published online by Cambridge University Press:  17 April 2009

Peter E. Kloeden
Peter E. Kloeden
Affiliation:
Department of Mathematics, Monash University, Clayton, Victoria.
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Abstract

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A continuous function mapping a compact interval of the real line into itself is called chaotic if the difference equation defined in terms of it behaves chaotically in the sense of Li and Yorke. The set of all such chaotic functions is shown to toe a dense subset of the space of continuous mappings of that interval into itself with the max norm. This result indicates the structural instability of nonchaotic difference equations with respect to chaotic behaviour.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 15 , Issue 3 , December 1976 , pp. 371 - 379
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Barna, Béla, “Über die Iteration reeller Funktionen I”, Publ. Math. Debrecen 7 (1960), 16–40.Google Scholar
[2]Barna, Béla, “Über die Iteration reeller Funktionen II”, Publ. Math. Debrecen 13 (1966), 169–172.CrossRefGoogle Scholar
[3]Barna, Béla, “Über die Iteration reeller Funktionen III”, Publ. Math. Debrecen 22 (1975), 269–278.Google Scholar
[4]Berg, Lothar, “Über irregulare Iterationsfolgen”, Publ. Math. Debrecen 17 (1970), 111–115.Google Scholar
[5]Coppel, W.A., “The solution of equations by iteration”, Proc. Cambridge Philos. Soc. 51 (1955), 41–43.CrossRefGoogle Scholar
[6]Li, Tien-yien and Yorke, James A., “Period three implies chaos”, Amer. Math. Monthly 82 (1975), 985–992.Google Scholar
[7]Lorenz, Edward N., “Deterministic nonperiodic flow”, J. Atmospheric Sci. 20 (1963), 130–141.Google Scholar
[8]Lorenz, Edward N., “The problem of deducing the climate from the governing equations”, Tellus 16 (1964), 1–11.Google Scholar
[9]Kloeden, Peter, Deakin, Michael A.B., Tirkel, A., “A precise definition of chaos”, Nature (to appear).Google Scholar
[10]May, Robert M., “Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos”, Science 186 (1974), 645–647.CrossRefGoogle ScholarPubMed
[11]Day, Robert M., “Biological populations obeying difference equations: stable points, stable cycles, and chaos”, J. Theoret. Biol. 51 (1975), 511–524.Google Scholar
[12]Шарковский, А.Н., [A.M. Šarkovskiiˇ], “Сосуществование циклов непрерывного преобраэованид прдмой в себя” [Coexistence of cycles of a continuous transformation of a line into itself], Ukrain. Mat. Z. 16 (1964), no. 1, 61–71.Google ScholarPubMed
[13] А.Н Шарковский, [A.N. Šarkovskiiˇ], “О циклах и с труктуре неприрывного отображения” [On cycles and the structure of a continuous transformation], Ukrain. Mat. Z. 17 (1965), no. 3, 104–111.Google Scholar