Distributional chaos for flows

Yunhua Zhou

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 2, page 475-480
  • ISSN: 0011-4642

Abstract

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Schweizer and Smítal introduced the distributional chaos for continuous maps of the interval in B. Schweizer, J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval. Trans. Amer. Math. Soc. 344 (1994), 737–854. In this paper, we discuss the distributional chaos DC1–DC3 for flows on compact metric spaces. We prove that both the distributional chaos DC1 and DC2 of a flow are equivalent to the time-1 maps and so some properties of DC1 and DC2 for discrete systems also hold for flows. However, we prove that DC2 and DC3 are not invariants of equivalent flows although DC2 is a topological conjugacy invariant in discrete case.

How to cite

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Zhou, Yunhua. "Distributional chaos for flows." Czechoslovak Mathematical Journal 63.2 (2013): 475-480. <http://eudml.org/doc/260593>.

@article{Zhou2013,
abstract = {Schweizer and Smítal introduced the distributional chaos for continuous maps of the interval in B. Schweizer, J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval. Trans. Amer. Math. Soc. 344 (1994), 737–854. In this paper, we discuss the distributional chaos DC1–DC3 for flows on compact metric spaces. We prove that both the distributional chaos DC1 and DC2 of a flow are equivalent to the time-1 maps and so some properties of DC1 and DC2 for discrete systems also hold for flows. However, we prove that DC2 and DC3 are not invariants of equivalent flows although DC2 is a topological conjugacy invariant in discrete case.},
author = {Zhou, Yunhua},
journal = {Czechoslovak Mathematical Journal},
keywords = {distributional chaos; flow; invariant; distributional chaos; continuous flow; flow equivalence},
language = {eng},
number = {2},
pages = {475-480},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Distributional chaos for flows},
url = {http://eudml.org/doc/260593},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Zhou, Yunhua
TI - Distributional chaos for flows
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 2
SP - 475
EP - 480
AB - Schweizer and Smítal introduced the distributional chaos for continuous maps of the interval in B. Schweizer, J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval. Trans. Amer. Math. Soc. 344 (1994), 737–854. In this paper, we discuss the distributional chaos DC1–DC3 for flows on compact metric spaces. We prove that both the distributional chaos DC1 and DC2 of a flow are equivalent to the time-1 maps and so some properties of DC1 and DC2 for discrete systems also hold for flows. However, we prove that DC2 and DC3 are not invariants of equivalent flows although DC2 is a topological conjugacy invariant in discrete case.
LA - eng
KW - distributional chaos; flow; invariant; distributional chaos; continuous flow; flow equivalence
UR - http://eudml.org/doc/260593
ER -

References

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  1. Balibrea, F., Smítal, J., Štefánková, M., The three versions of distributional chaos, Chaos Solitons Fractals 23 (2005), 1581-1583. (2005) Zbl1069.37013MR2101573
  2. Cao, Y., 10.1088/0951-7715/16/4/316, Nonlinearity 16 (2003), 1473-1479. (2003) Zbl1053.37014MR1986306DOI10.1088/0951-7715/16/4/316
  3. Downarowicz, T., Positive topological entropy implies chaos DC2, Arxiv.org/abs/1110.5201v1. 
  4. Schweizer, B., Smítal, J., 10.1090/S0002-9947-1994-1227094-X, Trans. Am. Math. Soc. 344 (1994), 737-754. (1994) Zbl0812.58062MR1227094DOI10.1090/S0002-9947-1994-1227094-X
  5. Smítal, J., Štefánková, M., 10.1016/j.chaos.2003.12.105, Chaos Solitons Fractals 21 (2004), 1125-1128. (2004) Zbl1060.37037MR2047330DOI10.1016/j.chaos.2003.12.105
  6. Sun, W., Young, T., Zhou, Y., 10.1090/S0002-9947-08-04743-0, Trans. Am. Math. Soc. 361 (2009), 3071-3082. (2009) Zbl1172.37002MR2485418DOI10.1090/S0002-9947-08-04743-0

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