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Journal of Dynamical Systems and Geometric Theories cover
Open Access ·Peer-reviewed·ISSN (Online): 2169-0057·ISSN (Print): 1726-037X

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Half-Yearly Journal: Publishes research on dynamical systems and geometry, including Random Dynamical Systems, hyperbolic behaviour, complex geometry and Ergodic theory.

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Open Access Original Articles

A Note on Devaney’s Definition of Chaos

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pp. 23–26Vol. 2Issue 1January 2004DOI: 10.1080/1726037X.2004.10698476XML
Received:
29 Sep 2003
Published Online:
03 Jun 2013
Article type:
Original Articles
Language:
EN
Article no.:
1726037X.2004.10698476
Pages:
23–26

Abstract

Devaney [5] defines a function to be chaotic if it satisfies three conditions: transitivity, having dense set of periodic points and sensitive dependence on initial conditions. Banks et al [2] prove that if the function is continuous then the third condition is implied from the first two and therefore it is redundant. However, if the function is not assumed to be continuous, then it is not known if the third condition is redundant or not.

Keywords

References

  1. Assaf, D and Gadbois, S.1992 . Definition of Chaos . , 99 ( 865 )
  2. Banks, J, Brooks, J, Cairns, G, Davis, G and Stacey, P.1992 . On Devaney’s Definition of Chaos . , 99 ( 332–334 )
  3. Block, LS and Coppel, WA. 1992 . , Vol. 1513 , Springer-Verlag . Lecture Notes in Mathematics
  4. Crannell, A.1995 . The Role of Transitivity in Devaney’s. Definition of Chaos . , 102 ( 788–793 )
  5. Devaney, RL.1989 . , 2nd , Addison-Wesley .
  6. Touhey, P.1997 . Yet Another Definition of Chaos . , 104 ( 411–414 )
  7. Vellekoop, M and Berglund, R.1994 . On Intervals, Transitivity=Chaos . , 101 ( 353–355 )
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