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

Discrete and Continuous Dynamical Systems

Volumn 19, Issue 1, 2007, Pages 103-119

  Lin, Wei ^a   Wu, Jianhong ^b   Chen, Guanrong ^c  

^a FUDAN UNIVERSITY   (China)

^b YORK UNIVERSITY   (Canada)

^c CITY UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Chaotic dynamics; Shift operator; Snap back repeller

Indexed keywords

EID: 34748881805     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2007.19.103     Document Type: Article

References (23)

1. R. Bowen, "On Axiom A Diffeomorphisms," Providence, RI: Amer. Math. Soc., 1978.
2. G. Chen, S. Hsu and J. Zhou, Snap-back revellers as a cause of chaotic vibration of the wave equation with a Van der Pol boundary condition and energy injection at the middle of the span, J. Math. Phys., 39 (1998), 6459-6489.
3. G. Chen, S. Hsu and J. Zhou, Nonisotropic spatiotemporal chaotic vibration of the wave equation due to mixing energy transport and a van der Pol boundary condition, Int. J. Bifurcation and Chaos, 12 (2002), 535-559.
4. G. Chen and D. Lai, Feedback anticontrol of chaos, Int. J. Bifurcation and Chaos, 8 (1998), 1585-1590.
5. L. Chen and K. Aihara, Chaos and asymptotical stability in discrete-time neural networks, Phys. D, 104 (1997), 286-325.
6. L. Chen and K. Aihara, Strange attractors in chaotic neural networks, IEEE Trans. on Circ and Syst. I: Fundamental Theory and Applications, 47 (2000), 1455-1468.
7. C. Li and G. Chen, On the Marotto-Li-Chen theorem and its application to chaotification of multi-dimensional discrete dynamical systems, Chaos, Solitons and Fractals, 18 (2003), 807-817.
8. R. L. Devaney, "An Introduction to Chaotic Dynamical Systems," 2nd ed. New York: Addison-Wesley, 1989.
9. J. Kennedy, S. Koçak and J. A. Yorke, Chaos lemma, American Mathematical Monthly, 108 (2001), 411-423.
10. J. Kennedy and J. A. Yorke, Topological horseshoes, Transaction of the Amer. Math. Soc., 353 (2001), 2513-2530.
11. T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992.
12. W. Lin, "Some Theoretical Problems in Complex Systems and Their Applications," PhD. dissertation, Inst. of Math., Fudan Univ., Shanghai, China, 2002.c
13. W. Lin and G. Chen, Heteroclinical revellers imply chaos, International Journal of Bifurcation and Chaos, 16 (2006), 1471-1489.
14. W. Lin, J. Ruan and W. Zhao, On the mathematical clarification of the snap-back-repeller in high-dimensional systems and chaos in a discrete neural network model, Int. J. Bifurcation Chaos, 12 (2002), 1129-1139.
15. W. Lin and J. Ruan, Chaotic dynamics of an integrate-and-fire circuit with periodic pulse-train input, IEEE Trans. on Circ. and Syst. I: Fundamental Theory and Applications, 50 (2003), 686-693.
16. D. Lind and B. Marcus, "An Introduction to Symbolic Dynamics and Coding," Cambridge University Press, 1995.
17. n, J. Math. Anal. Appl., 63 (1978), 199-223.
18. F. R. Marotto, On redefining a snap-back repeller, Chaos, Solitons and Fractals, 25 (2005), 25-28.
19. W. de Melo and S.J. Van Strien, "One-Dimensional Dynamics," New York: Springer-Verlag, 1993.
20. C. Robinson, "Dynamical Systems: Stability, Symbolic Dynamics, and Chaos," 2nd ed. CRC Press, 1998.
21. Y. Shi and G. Chen, Discrete chaos in Banach spaces, Science in China, Series A, 34 (2004), 595-609.
22. Y. Shi and G. Chen, Chaos of discrete dynamical systems in complete metric spaces, Chaos, Solitons and Fractals, 22 (2004), 555-571.
23. H. Zhang and G. Chen, Single-input multi-output state-feedback chaotification of general discrete systems, Int. J. Bifurcation and Chaos, 14 (2004), 3317-3323.