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An algorithm for counting the number of periodic points of a family of polynomials | ||
| Journal of Discrete Mathematics and Its Applications | ||
| دوره 9، شماره 4، اسفند 2024، صفحه 249-267 اصل مقاله (366.13 K) | ||
| نوع مقاله: Full Length Article | ||
| شناسه دیجیتال (DOI): 10.22061/jdma.2024.11165.1084 | ||
| نویسندگان | ||
| Monireh Akbari* 1؛ Maryam Rabii2 | ||
| 1Department of Mathematics, Faculty of Basic Sciences, Shahid Rajaee Teacher Training University, Tehran, Iran | ||
| 2Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University,Tehran, Iran | ||
| تاریخ دریافت: 03 آبان 1403، تاریخ بازنگری: 15 آبان 1403، تاریخ پذیرش: 29 آبان 1403 | ||
| چکیده | ||
| In this paper we consider the family fa(x) = axd(x − 1) + x when a < 0 is a real number and d ≥ 2 is an even integer. The function fa has a unique positive critical point. By decreasing the parameter a, the behavior of the orbit of this critical point changes. In this paper we consider two cases. In the first case the orbit of the positive critical point converges to 0 and in the second case the positive critical point is mapped to a repelling periodic point of period 2. In each case we give a recursive formula to determine the number of the periodic points of fa. Also, in each case we introduce an invariant set on which fa is chaotic. We employ conjugacy map and symbolic dynamics in our investigations. | ||
| کلیدواژه‌ها | ||
| Cantor set؛ chaos؛ conjugacy؛ periodic points؛ symbolic dynamics | ||
| مراجع | ||
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[1] M. Akbari, M. Rabii, Real cubic polynomials with a fixed point of multiplicity two, Indagationes Mathematicae 26 (2015) 64–74. https://doi.org/10.1016/j.indag.2014.06.001 [2] H. Bai-lin, Elementary Symbolic Dynamics and Chaos in Dissipative Systems, World Scientific, 1989. https://doi.org/10.1142/0674 [3] S. Baldwin, Continuous itinerary functions and dendrite maps, Topology and its Applications 154 (2007) 2889–2938. https://doi.org/10.1016/j.topol.2007.04.001 [4] S. Baldwin, Julia sets and periodic kneading sequences, Journal of Fixed Point Theory and Applications 7 (2010) 201–222 . https://doi.org/10.1007/s11784-010-0007-y [5] J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, On Devaney’s definition of chaos, The American Mathematical Monthly 99 (1992) 332–334. https://www.jstor.org/stable/2324899 [6] A. F. Beardon, Iteration of Rational Functions, Complex Analytic Dynamical Systems, SpringerVerlag, 1991. https://doi.org/10.1007/978-1-4612-4422-6 [7] H. Bruin, D. Schleicher, Symbolic Dynamics of Quadratic Polynomials, Institut Mittag-Leffler, report no. 7, 2001. https://www.mat.univie.ac.at/ bruin/talks/TreesBook.pdf [8] W. de Melo, S. Van Strien, One-Dimensional Dynamics, Springer-Verlag, Berlin, 1993. https://doi.org/10.1007/978-3-642-78043-1 [9] R. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd edition, Westview, 2003. https://doi.org/10.4324/9780429502309 [10] R. Devaney, D. Look, Symbolic dynamics for a Sierpinski curve Julia set, Journal of Difference Equations and Applications 11(7) (2005) 581–596 . https://doi.org/10.1080/10236190412331334473 [11] D. Lind, B. Marcus, An introduction to symbolic dynamics and coding, Cambridge University, 1995. https://doi.org/10.1017/cbo9780511626302 [12] S. Elaydi, Discrete Chaos, 2nd edition, Chapman and Hall/CRC, 2008. https://doi.org/10.1201/9781420011043 [13] J. Milnor, Dynamics in One Complex Variable, 3rd edition, Princeton University Press, 2006. https://doi.org/10.1007/978-3-663-08092-3 [14] S. Li, ω chaos and topological entropy, Transactions of the American Mathematical Society 339(1) (1993) 243-249. https://doi.org/10.2307/2154217 | ||
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