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A generalization of a theorem of marotto

Published online by Cambridge University Press:  22 January 2016

Kenichi Shiraiwa  and
Masahiro Kurata
Kenichi Shiraiwa
Affiliation:
Department of Mathematics, College of General Education, Nagoya University
Masahiro Kurata
Affiliation:
Department of Mathematics, Nagoya Institute of Technology
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In 1975, Li and Yorke [3] found the following fact. Let f: I→ I be a continuous map of the compact interval I of the real line R into itself. If f has a periodic point of minimal period three, then f exhibits chaotic behavior. The above result is generalized by F.R. Marotto [4] in 1978 for the multi-dimensional case as follows. Let f: Rn → Rn be a differentiate map of the n-dimensional Euclidean space Rn (n ≧ 1) into itself. If f has a snap-back repeller, then f exhibits chaotic behavior.

In this paper, we give a generalization of the above theorem of Marotto. Our theorem can also be regarded as a generalization of the Smale’s results on the transversal homoclinic point of a diffeomorphism.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 82 , June 1981 , pp. 83 - 97
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1981

References

[1] Hartman, P., Ordinary differential equations, John Wiley and Sons, Inc. (1964), p. 245, Lemma 8.1.Google Scholar
[2] Kurata, M., Hartman’s theorem for hyperbolic sets, Nagoya Math. J., 67 (1977), 41–52.Google Scholar
[3] Li, T. Y. and Yorke, J. A., Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985–992.CrossRefGoogle Scholar
[4] Marotto, F. R., Snap-back repellers imply chaos in Rn , J. Math. Analysis and Appl., 63 (1978), 199–223.Google Scholar
[5] Palis, J., On Morse-Smale dynamical systems, Topology, 8 (1969), 385–404.Google Scholar
[6] Smale, S., Diffeomorphisms with many periodic points, Differential and Combinatorial Topology, Princeton University Press (1965), 63–80.Google Scholar