On the maximal invariant set for the map X² - 2 restricted to intervals
DOI:
https://doi.org/10.22199/issn.0717-6279-2021-02-0018Keywords:
Chaos, Tent maps, Maximal invariant, Unimodal maps, Quadratic mapsAbstract
In this paper, we study the maximal invariant set of a quadratic family related to a class of unimodal maps. This family is very important and have direct application in many branches of science. In particular, we characterize when the maximal invariant of f(x) = x2 − 2 (restricted to an interval) has a chaotic behavior.
References
L. S. Block and W. A. Coppel, Dynamics in one dimension. Berlin: Springer, 1992, doi: 10.1007/BFb0084762
R. L. Kraft, “Chaos, cantor sets, and hyperbolicity for the logistic maps”, The american mathematical monthly, vol. 106, no. 5, pp. 400–408, 1999, doi: 10.1080/00029890.1999.12005062
S. Grossmann and S. Thomae, “Invariant distributions and stationary correlation functions of one-dimensional discrete processes”, Zeitschrift für naturforschung A, vol. 32, no. 12, pp. 1353–1363, 1977, doi: 10.1515/zna-1977-1204
A. Lasota and J. A. Yorke, “On the existence of invariant measures for piecewise monotonic transformations”, Transactions of the American Mathematical Society, vol. 186, pp. 481–488, 1973, doi: 10.1090/S0002-9947-1973-0335758-1
A. Lasota and M. M. Mackey, Chaos, fractals, and noise: stochastic aspects of dynamics. New York, NY: Springer, 1994, doi: 10.1007/978-1-4612-4286-4
W. de Melo and S. van Strien, One-dimensional dynamics. Berlin: Springer, 1993, doi: 10.1007/978-3-642-78043-1
R. M. da. Silva, C. Manchein, and M. W. Beims, “Controlling intermediate dynamics in a family of quadratic maps,” Chaos: an interdisciplinary journal of nonlinear science, vol. 27, no. 10, Art ID. 103101, 2017, doi: 10.1063/1.4985331
C. G. Moreira, “Maximal invariant sets for restrictions of tent and unimodal maps,” Qualitative theory of dynamical systems, vol. 2, no. 2, pp. 385–398, 2001, doi: 10.1007/BF02969348
S. V. Raković and M. Fiacchini, “Invariant approximations of the maximal invariant set or ‘Encircling the Square’”, IFAC proceedings volumes, vol. 41, no. 2, pp. 6377–6382, 2008, doi: 10.3182/20080706-5-KR-1001.01076
Published
How to Cite
Issue
Section
Copyright (c) 2021 Dušan Bednařík, Diego Marques, Carlos Gustavo Tamm de Araujo Moreira, Pavel Trojovský
This work is licensed under a Creative Commons Attribution 4.0 International License.
-
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.
