March  2007, 19(1): 103-119. doi: 10.3934/dcds.2007.19.103

Generalized snap-back repeller and semi-conjugacy to shift operators of piecewise continuous transformations

1. 

Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, School of Mathematical Sciences, Fudan University, Shanghai 200433, China

2. 

Department of Mathematics and Statistics, York University, Toronto, Canada M3J 1P3

3. 

Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong S.A.R., China

Received  August 2005 Revised  May 2007 Published  June 2007

In this paper, we attempt to clarify an open problem related to a generalization of the snap-back repeller. Constructing a semi-conjugacy from the finite product of a transformation $f:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ on an invariant set $\Lambda$ to a sub-shift of the finite type on a $w$-symbolic space, we show that the corresponding transformation associated with the generalized snap-back repeller on $\mathbb{R}^{n}$ exhibits chaotic dynamics in the sense of having a positive topological entropy. The argument leading to this conclusion also shows that a certain kind of degenerate transformations, admitting a point in the unstable manifold of a repeller mapping back to the repeller, have positive topological entropies on the orbits of their invariant sets. Furthermore, we present two feasible sufficient conditions for obtaining an unstable manifold. Finally, we provide two illustrative examples to show that chaotic degenerate transformations are omnipresent.
Citation: Wei Lin, Jianhong Wu, Guanrong Chen. Generalized snap-back repeller and semi-conjugacy to shift operators of piecewise continuous transformations. Discrete and Continuous Dynamical Systems, 2007, 19 (1) : 103-119. doi: 10.3934/dcds.2007.19.103
[1]

James Kingsbery, Alex Levin, Anatoly Preygel, Cesar E. Silva. Dynamics of the $p$-adic shift and applications. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 209-218. doi: 10.3934/dcds.2011.30.209

[2]

Sébastien Guisset. Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations. Kinetic and Related Models, 2020, 13 (4) : 739-758. doi: 10.3934/krm.2020025

[3]

Proscovia Namayanja. Chaotic dynamics in a transport equation on a network. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3415-3426. doi: 10.3934/dcdsb.2018283

[4]

Cezar Joiţa, William O. Nowell, Pantelimon Stănică. Chaotic dynamics of some rational maps. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 363-375. doi: 10.3934/dcds.2005.12.363

[5]

Helge Krüger. Asymptotic of gaps at small coupling and applications of the skew-shift Schrödinger operator. Conference Publications, 2011, 2011 (Special) : 874-880. doi: 10.3934/proc.2011.2011.874

[6]

Samira Amraoui, Didier Auroux, Jacques Blum, Emmanuel Cosme. Back-and-forth nudging for the quasi-geostrophic ocean dynamics with altimetry: Theoretical convergence study and numerical experiments with the future SWOT observations. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022058

[7]

J. Leonel Rocha, Danièle Fournier-Prunaret, Abdel-Kaddous Taha. Strong and weak Allee effects and chaotic dynamics in Richards' growths. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2397-2425. doi: 10.3934/dcdsb.2013.18.2397

[8]

Roman Srzednicki. A theorem on chaotic dynamics and its application to differential delay equations. Conference Publications, 2001, 2001 (Special) : 362-365. doi: 10.3934/proc.2001.2001.362

[9]

Guihong Fan, Gail S. K. Wolkowicz. Chaotic dynamics in a simple predator-prey model with discrete delay. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 191-216. doi: 10.3934/dcdsb.2020263

[10]

Alfonso Ruiz Herrera. Paradoxical phenomena and chaotic dynamics in epidemic models subject to vaccination. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2533-2548. doi: 10.3934/cpaa.2020111

[11]

Chao Wang, Dingbian Qian, Qihuai Liu. Impact oscillators of Hill's type with indefinite weight: Periodic and chaotic dynamics. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2305-2328. doi: 10.3934/dcds.2016.36.2305

[12]

Emile Franc Doungmo Goufo, Melusi Khumalo, Patrick M. Tchepmo Djomegni. Perturbations of Hindmarsh-Rose neuron dynamics by fractional operators: Bifurcation, firing and chaotic bursts. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 663-682. doi: 10.3934/dcdss.2020036

[13]

Marc Bocquet, Julien Brajard, Alberto Carrassi, Laurent Bertino. Bayesian inference of chaotic dynamics by merging data assimilation, machine learning and expectation-maximization. Foundations of Data Science, 2020, 2 (1) : 55-80. doi: 10.3934/fods.2020004

[14]

Anastasiia Panchuk, Frank Westerhoff. Speculative behavior and chaotic asset price dynamics: On the emergence of a bandcount accretion bifurcation structure. Discrete and Continuous Dynamical Systems - B, 2021, 26 (11) : 5941-5964. doi: 10.3934/dcdsb.2021117

[15]

Van Cyr, John Franks, Bryna Kra, Samuel Petite. Distortion and the automorphism group of a shift. Journal of Modern Dynamics, 2018, 13: 147-161. doi: 10.3934/jmd.2018015

[16]

Daniel Gonçalves, Marcelo Sobottka. Continuous shift commuting maps between ultragraph shift spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1033-1048. doi: 10.3934/dcds.2019043

[17]

Michael Schraudner. Projectional entropy and the electrical wire shift. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 333-346. doi: 10.3934/dcds.2010.26.333

[18]

Michael Baake, John A. G. Roberts, Reem Yassawi. Reversing and extended symmetries of shift spaces. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 835-866. doi: 10.3934/dcds.2018036

[19]

Christian Wolf. A shift map with a discontinuous entropy function. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 319-329. doi: 10.3934/dcds.2020012

[20]

Haripriya Barman, Magfura Pervin, Sankar Kumar Roy, Gerhard-Wilhelm Weber. Back-ordered inventory model with inflation in a cloudy-fuzzy environment. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1913-1941. doi: 10.3934/jimo.2020052

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]