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 & Continuous Dynamical Systems - A, 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.   Google Scholar

[2]

L. Block and W. A. Coppel, Dynamics in One Dimension,, Lecture Notes in Mathematics 1513, (1513).   Google Scholar

[3]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems,, Second edition. Addison-Wesley Studies in Nonlinearity. Addison-Wesley Publishing Company, (1989).   Google Scholar

[4]

B. Hasselblatt and A. Katok, A First Course in Dynamics: With a Panorama of Recent Developments,, Cambridge University Press, (2003).   Google Scholar

[5]

R. L. Kraft, Chaos, Cantor sets, and hyperbolicity for the logistic maps,, Amer. Math. Monthly, 106 (1999), 400.  doi: 10.2307/2589144.  Google Scholar

[6]

T.-Y. Li and J. Yorke, Period three implies chaos,, Amer. Math. Monthly, 82 (1975), 985.  doi: 10.2307/2318254.  Google Scholar

[7]

A. N. Sharkovsky, Co-existence of cycles of a continuous mapping of the line into itself,, Ukrain. Math. J., 16 (1964), 61.   Google Scholar

[8]

D. Singer, Stable orbits and bifurcation of maps of the interval,, SIAM J. Appl. Math., 35 (1978), 260.  doi: 10.1137/0135020.  Google Scholar

[9]

X. Zhang, Y. Shi and G. Chen, Some properties of coupled-expanding maps in compact sets,, Proc. Amer. Math. Soc., 141 (2013), 585.  doi: 10.1090/S0002-9939-2012-11339-5.  Google Scholar

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.   Google Scholar

[2]

L. Block and W. A. Coppel, Dynamics in One Dimension,, Lecture Notes in Mathematics 1513, (1513).   Google Scholar

[3]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems,, Second edition. Addison-Wesley Studies in Nonlinearity. Addison-Wesley Publishing Company, (1989).   Google Scholar

[4]

B. Hasselblatt and A. Katok, A First Course in Dynamics: With a Panorama of Recent Developments,, Cambridge University Press, (2003).   Google Scholar

[5]

R. L. Kraft, Chaos, Cantor sets, and hyperbolicity for the logistic maps,, Amer. Math. Monthly, 106 (1999), 400.  doi: 10.2307/2589144.  Google Scholar

[6]

T.-Y. Li and J. Yorke, Period three implies chaos,, Amer. Math. Monthly, 82 (1975), 985.  doi: 10.2307/2318254.  Google Scholar

[7]

A. N. Sharkovsky, Co-existence of cycles of a continuous mapping of the line into itself,, Ukrain. Math. J., 16 (1964), 61.   Google Scholar

[8]

D. Singer, Stable orbits and bifurcation of maps of the interval,, SIAM J. Appl. Math., 35 (1978), 260.  doi: 10.1137/0135020.  Google Scholar

[9]

X. Zhang, Y. Shi and G. Chen, Some properties of coupled-expanding maps in compact sets,, Proc. Amer. Math. Soc., 141 (2013), 585.  doi: 10.1090/S0002-9939-2012-11339-5.  Google Scholar

[1]

Benjamin Webb. Dynamics of functions with an eventual negative Schwarzian derivative. Discrete & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 2015, 35 (2) : 653-668. doi: 10.3934/dcds.2015.35.653

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[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]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

Plamen Stefanov, Gunther Uhlmann, Andras Vasy. On the stable recovery of a metric from the hyperbolic DN map with incomplete data. Inverse Problems & Imaging, 2016, 10 (4) : 1141-1147. doi: 10.3934/ipi.2016035

[19]

Amadeu Delshams, Marian Gidea, Pablo Roldán. Transition map and shadowing lemma for normally hyperbolic invariant manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1089-1112. doi: 10.3934/dcds.2013.33.1089

[20]

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

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]