Mathematical  Culture and Thought

Mathematical Culture and Thought

An Overview of the Concept of Chaos in Discrete Dynamical Systems

Document Type : Review

Authors
1 Department of Mathematics, Shahid Rajaee Teacher Training University, Iran
2 Department of Mathematics, Alzahra University, Iran
Abstract
Our aim in this paper is to introduce and study five common definitions of chaos in discrete
dynamical systems and compare them to each other on compact intervals.These five different
definitions that describe chaos from different points of view are Li-Yorke chaos, topological chaos,
ω-chaos, Block-Coppel chaos, and Devaney chaos.
Keywords

Subjects


 
[1]   Akbari, M.,   Rabii, M., Converse of  Sarkovskii's theorem, Mathematical Culture and Thought, 66 (2020), 115-133. [in Persian]
[2] Aulbach, B., Kieninger, B., On Three Definitions of Chaos, Nonlinear Dyn. Syst. Theory, 1(1)
(2001), 23-37.
[3] Assaf, D., Gadbois, S., Definitions of chaos, Amer. Math. Monthly, 99 (1992), 865.
[4] Banks, J., Brooks, J., Carins, G., Davis, G., Stacey, P., On Devaney’s definition of chaos, Amer.
Math. Monthly, 99 (1992), 332-334.
[5] Birkhoff, G. D., Dynamical Systems, Amer. Math. Soc., Providence, RI, 1927.
[6] Birkhoff, G. D., Collected Mathematical Papers, Vols. 1, 2, 3, Literary Licensing, LLC, New York,
1950.
[7] Blanchard, F., Topological chaos: What may this mean?, J. Difference Equ. Appl., 15(1) (2009),
23-46.
[8] Block, L. S., Coppel, W. A., Dynamics in One Dimension, Springer-Verlag, New York, 1992.
[9] Collet, P., Eckmann, J. P., Iterated Maps on the Interval as Dynamical Systems, Progress in Physics,
1, Birkhäuser, Basel, 1980.
[10] Devaney, R., An Introduction to Chaotic Dynamical Systems, Benjamin/Cummings, Menlo Park,
Calif., 1986.
[11] Elaydi, S. N., Discrete Chaos, With Applications in Science and Engineering, 2nd ed., Chapman &
Hall/CRC, Boca Raton, 2007.
[12] Furstenberg, H., Disjointness in ergodic theory, minimal sets and a problem in diophantine approximation,
Mathematical Systems Theory, 1 (1967), 1-49.
[13] Glasner, E., Weiss, B., Sensitive dependence on initial conditions, Nonlinearity, 6 (1993), 1067-
1075.
[14] Li, S., ω-Chaos and topological entropy, Trans. Amer. Math. Soc., 339 (1) (1993), 243–249
[15] Li, T. Y., Yorke, J., Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992.
[16]   Rahimi, M.,   Mirzaei, M., Ergodic theory: dynamical systems from functional analysis view point, Mathematical Culture and Thought, 64 (2019), 59-78. [in Persian]
[17]   Razminia, A., A note on limit sets in dynamical systems, Mathematical Culture and Thought, 66 (2015), 49-68. [in Persian]
 
[18] Šharkovskii, A. N., Co-existence of the cycles of a continuous mapping of the line into itself (Russian),
Ukrainian Math. J., 16 (1964), 61-71.
[19] Šharkovskii, A. N., About continuous maps on the set of co-limit points, Proc. Acad. Sci. Ukraine,
(1965), 1407-1410.
[20] Šharkovskii, A. N., Behavior of mappings in the neighborhood of an attracting set, Ukrainian Math.
J., 18 (1966), 60-83.
[21] Šharkovskii, A. N., Kolyada, S. F., Sivak, A. G., Fedorenko, V. V., Dynamics of One- dimensional
Mappings(Russian), Naukova Dumka, Kiev, 1989.
[22] Silvermann, S., On maps with dense orbits and the definition of chaos, Rocky Mountain J. Math.,
22(1) (1992), 353-375.
[23] Smítal, J., Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc., 297 (1986),
269-282.
[24] Vellekoop, M., Berglund, R., On intervals, Transitivity=Chaos, Amer. Math. Monthly, 101 (1994),
353-355.

  • Receive Date 18 December 2021
  • Revise Date 08 June 2022
  • Accept Date 27 June 2022
  • Publish Date 21 January 2023