American Institute of Mathematical Sciences

May  2014, 34(5): 1933-1949. doi: 10.3934/dcds.2014.34.1933

Period 3 and chaos for unimodal maps

 1 Department of Applied Mathematics, Chung Yuan Christian University, Chungli, Taiwan, Taiwan 2 Department of Financial and Computational Mathematics, Providence University, Taichung, Taiwan

Received  January 2013 Revised  July 2013 Published  October 2013

In this paper we study unimodal maps on the closed unit interval, which have a stable period 3 orbit and an unstable period 3 orbit, and give conditions under which all points in the open unit interval are either asymptotic to the stable period 3 orbit or land after a finite time on an invariant Cantor set $\Lambda$ on which the dynamics is conjugate to a subshift of finite type and is, in fact, chaotic. For the particular value of $\mu=3.839$, Devaney [3], following ideas of Smale and Williams, shows that the logistic map $f(x)=\mu x(1-x)$ has this property. In this case the stable and unstable period 3 orbits appear when $\mu=\mu_0=1+\sqrt{8}$. We use our theorem to show that the property holds for all values of $\mu>\mu_0$ for which the stable period 3 orbit persists.
Citation: Kaijen Cheng, Kenneth Palmer, Yuh-Jenn Wu. Period 3 and chaos for unimodal maps. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1933-1949. doi: 10.3934/dcds.2014.34.1933
References:
 [1] B. Aulbach and B. Kieninger, An elementary proof for hyperbolicity and chaos of the logistic maps, J. Difference Eqns. Appl., 10 (2004), 1243-1250. [2] L. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Mathematics 1513, Springer-Verlag, New York, 1992. [3] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Second edition. Addison-Wesley Studies in Nonlinearity. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. [4] B. Hasselblatt and A. Katok, A First Course in Dynamics: With a Panorama of Recent Developments, Cambridge University Press, New York, 2003. [5] R. L. Kraft, Chaos, Cantor sets, and hyperbolicity for the logistic maps, Amer. Math. Monthly, 106 (1999), 400-408. doi: 10.2307/2589144. [6] T.-Y. Li and J. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992. doi: 10.2307/2318254. [7] A. N. Sharkovsky, Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Math. J., 16 (1964), 61-71. [8] D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267. doi: 10.1137/0135020. [9] X. Zhang, Y. Shi and G. Chen, Some properties of coupled-expanding maps in compact sets, Proc. Amer. Math. Soc., 141 (2013), 585-595. doi: 10.1090/S0002-9939-2012-11339-5.

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References:
 [1] B. Aulbach and B. Kieninger, An elementary proof for hyperbolicity and chaos of the logistic maps, J. Difference Eqns. Appl., 10 (2004), 1243-1250. [2] L. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Mathematics 1513, Springer-Verlag, New York, 1992. [3] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Second edition. Addison-Wesley Studies in Nonlinearity. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. [4] B. Hasselblatt and A. Katok, A First Course in Dynamics: With a Panorama of Recent Developments, Cambridge University Press, New York, 2003. [5] R. L. Kraft, Chaos, Cantor sets, and hyperbolicity for the logistic maps, Amer. Math. Monthly, 106 (1999), 400-408. doi: 10.2307/2589144. [6] T.-Y. Li and J. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992. doi: 10.2307/2318254. [7] A. N. Sharkovsky, Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Math. J., 16 (1964), 61-71. [8] D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267. doi: 10.1137/0135020. [9] X. Zhang, Y. Shi and G. Chen, Some properties of coupled-expanding maps in compact sets, Proc. Amer. Math. Soc., 141 (2013), 585-595. doi: 10.1090/S0002-9939-2012-11339-5.
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