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.

show all references

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.

[1]

Benjamin Webb. Dynamics of functions with an eventual negative Schwarzian derivative. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1393-1408. doi: 10.3934/dcds.2009.24.1393

[2]

Eduardo Liz, Manuel Pinto, Gonzalo Robledo, Sergei Trofimchuk, Victor Tkachenko. Wright type delay differential equations with negative Schwarzian. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 309-321. doi: 10.3934/dcds.2003.9.309

[3]

Denis Volk. Almost every interval translation map of three intervals is finite type. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2307-2314. doi: 10.3934/dcds.2014.34.2307

[4]

J. Alberto Conejero, Francisco Rodenas, Macarena Trujillo. Chaos for the Hyperbolic Bioheat Equation. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 653-668. doi: 10.3934/dcds.2015.35.653

[5]

Frédéric Faure. Prequantum chaos: Resonances of the prequantum cat map. Journal of Modern Dynamics, 2007, 1 (2) : 255-285. doi: 10.3934/jmd.2007.1.255

[6]

C. Bonanno, G. Menconi. Computational information for the logistic map at the chaos threshold. Discrete and Continuous Dynamical Systems - B, 2002, 2 (3) : 415-431. doi: 10.3934/dcdsb.2002.2.415

[7]

James W. Cannon, Mark H. Meilstrup, Andreas Zastrow. The period set of a map from the Cantor set to itself. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2667-2679. doi: 10.3934/dcds.2013.33.2667

[8]

Yi Yang, Robert J. Sacker. Periodic unimodal Allee maps, the semigroup property and the $\lambda$-Ricker map with Allee effect. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 589-606. doi: 10.3934/dcdsb.2014.19.589

[9]

Àngel Jorba, Pau Rabassa, Joan Carles Tatjer. Period doubling and reducibility in the quasi-periodically forced logistic map. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1507-1535. doi: 10.3934/dcdsb.2012.17.1507

[10]

Partha Sharathi Dutta, Soumitro Banerjee. Period increment cascades in a discontinuous map with square-root singularity. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 961-976. doi: 10.3934/dcdsb.2010.14.961

[11]

S. Jiménez, Pedro J. Zufiria. Characterizing chaos in a type of fractional Duffing's equation. Conference Publications, 2015, 2015 (special) : 660-669. doi: 10.3934/proc.2015.0660

[12]

Yue-Jun Peng, Yong-Fu Yang. Long-time behavior and stability of entropy solutions for linearly degenerate hyperbolic systems of rich type. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3683-3706. doi: 10.3934/dcds.2015.35.3683

[13]

Shujuan Lü, Chunbiao Gan, Baohua Wang, Linning Qian, Meisheng Li. Traveling wave solutions and its stability for 3D Ginzburg-Landau type equation. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 507-527. doi: 10.3934/dcdsb.2011.16.507

[14]

Jawad Al-Khal, Henk Bruin, Michael Jakobson. New examples of S-unimodal maps with a sigma-finite absolutely continuous invariant measure. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 35-61. doi: 10.3934/dcds.2008.22.35

[15]

Iuliana Oprea, Gerhard Dangelmayr. A period doubling route to spatiotemporal chaos in a system of Ginzburg-Landau equations for nematic electroconvection. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 273-296. doi: 10.3934/dcdsb.2018095

[16]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094

[17]

Jianlu Zhang. Coexistence of period 2 and 3 caustics for deformative nearly circular billiard maps. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6419-6440. doi: 10.3934/dcds.2019278

[18]

A. Crannell. A chaotic, non-mixing subshift. Conference Publications, 1998, 1998 (Special) : 195-202. doi: 10.3934/proc.1998.1998.195

[19]

Denis Gaidashev, Tomas Johnson. Dynamics of the universal area-preserving map associated with period-doubling: Stable sets. Journal of Modern Dynamics, 2009, 3 (4) : 555-587. doi: 10.3934/jmd.2009.3.555

[20]

Jiaxi Huang, Youde Wang, Lifeng Zhao. Equivariant Schrödinger map flow on two dimensional hyperbolic space. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4379-4425. doi: 10.3934/dcds.2020184

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (95)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]