American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Null controllability of a cascade system of parabolic-hyperbolic equations
  • DCDS Home
  • This Issue
  • Next Article
    Existence problem for the Kirchhoff type wave equation with a localized weakly nonlinear dissipation in exterior domains
February & March  2004, 11(2&3): 715-730. doi: 10.3934/dcds.2004.11.715

Attractors from one dimensional Lorenz-like maps

Youngna Choi 1,

1. 

Department of Mathematical Sciences, Montclair State University, Montclair, NJ 07043, United States

Received  November 2002 Revised  April 2004 Published  June 2004

In this paper we study the properties of expanding maps with a single discontinuity on a closed interval and the resultant dynamics. For such a map, there exists a compact invariant subset which shares a lot of common properties with classical attractors such as the topological transitivity of the restricted map and the density of the periodic points. The invariant set, with more conditions on the boundary, can be shown to have an isolating neighborhood, hence is a chaotic attractor in the strong sense. Not all such maps derive trapping regions, yet by perturbation, those non-attractors can be made to have a trapping region.
Keywords: Expanding maps, attractors., invariant sets.
Mathematics Subject Classification: 37E05, 37G3.
Citation: Youngna Choi. Attractors from one dimensional Lorenz-like maps. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 715-730. doi: 10.3934/dcds.2004.11.715
[1]

Antonio Pumariño, José Ángel Rodríguez, Enrique Vigil. Renormalizable Expanding Baker Maps: Coexistence of strange attractors. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1651-1678. doi: 10.3934/dcds.2017068
[2]

Tifei Qian, Zhihong Xia. Heteroclinic orbits and chaotic invariant sets for monotone twist maps. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 69-95. doi: 10.3934/dcds.2003.9.69
[3]

Xueting Tian, Paulo Varandas. Topological entropy of level sets of empirical measures for non-uniformly expanding maps. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5407-5431. doi: 10.3934/dcds.2017235
[4]

Antonio Pumariño, José Ángel Rodríguez, Enrique Vigil. Persistent two-dimensional strange attractors for a two-parameter family of Expanding Baker Maps. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 657-670. doi: 10.3934/dcdsb.2018201
[5]

Carlangelo Liverani. A footnote on expanding maps. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3741-3751. doi: 10.3934/dcds.2013.33.3741
[6]

Peter Haïssinsky, Kevin M. Pilgrim. An algebraic characterization of expanding Thurston maps. Journal of Modern Dynamics, 2012, 6 (4) : 451-476. doi: 10.3934/jmd.2012.6.451
[7]

Peter Haïssinsky, Kevin M. Pilgrim. Examples of coarse expanding conformal maps. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2403-2416. doi: 10.3934/dcds.2012.32.2403
[8]

José F. Alves. Stochastic behavior of asymptotically expanding maps. Conference Publications, 2001, 2001 (Special) : 14-21. doi: 10.3934/proc.2001.2001.14
[9]

Yushi Nakano, Shota Sakamoto. Spectra of expanding maps on Besov spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1779-1797. doi: 10.3934/dcds.2019077
[10]

Frank D. Grosshans, Jürgen Scheurle, Sebastian Walcher. Invariant sets forced by symmetry. Journal of Geometric Mechanics, 2012, 4 (3) : 271-296. doi: 10.3934/jgm.2012.4.271
[11]

Rafael De La Llave, Michael Shub, Carles Simó. Entropy estimates for a family of expanding maps of the circle. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 597-608. doi: 10.3934/dcdsb.2008.10.597
[12]

Michael Blank. Finite rank approximations of expanding maps with neutral singularities. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 749-762. doi: 10.3934/dcds.2008.21.749
[13]

Xu Zhang, Yuming Shi, Guanrong Chen. Coupled-expanding maps under small perturbations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1291-1307. doi: 10.3934/dcds.2011.29.1291
[14]

Viviane Baladi, Daniel Smania. Smooth deformations of piecewise expanding unimodal maps. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 685-703. doi: 10.3934/dcds.2009.23.685
[15]

Yong Fang. On smooth conjugacy of expanding maps in higher dimensions. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 687-697. doi: 10.3934/dcds.2011.30.687
[16]

Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917
[17]

Yiming Ding. Renormalization and $\alpha$-limit set for expanding Lorenz maps. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 979-999. doi: 10.3934/dcds.2011.29.979
[18]

Ralf Spatzier, Lei Yang. Exponential mixing and smooth classification of commuting expanding maps. Journal of Modern Dynamics, 2017, 11: 263-312. doi: 10.3934/jmd.2017012
[19]

Alena Erchenko. Flexibility of Lyapunov exponents for expanding circle maps. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2325-2342. doi: 10.3934/dcds.2019098
[20]

Zhihong Xia. Hyperbolic invariant sets with positive measures. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 811-818. doi: 10.3934/dcds.2006.15.811

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences