Generalized Hénon maps and smale horseshoes of new types

International Journal of Bifurcation and Chaos

Volumn 18, Issue 10, 2008, Pages 3029-3052

  Gonchenko, Sergey ^a   Li, Ming Chia ^b   Malkin, Mikhail ^a  

^a LOBACHEVSKY STATE UNIVERSITY OF NIZHNY NOVGOROD   (Russian Federation)

^b NATIONAL CHIAO TUNG UNIVERSITY   (Taiwan)

Author keywords

Half orientable horseshoe; H non map; Hyperbolic dynamics; Nonwandering set; Singular bifurcation; Smale horseshoe

Indexed keywords

BIFURCATION (MATHEMATICS);

HALF-ORIENTABLE HORSESHOE; HYPERBOLIC DYNAMICS; NONWANDERING SET; SINGULAR BIFURCATION; SMALE HORSESHOE;

CHAOS THEORY;

EID: 59049099319     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127408022238     Document Type: Article

References (29)

1. Afraimovich, V. S., Bykov, V. V. & Shilnikov, L. P. [1977] "On the origin and structure of the Lorenz attractor," Soviet Phys. Dokl. 22, 253-255.
2. Afraimovich, V. S., Bykov, V. V. & Shilnikov, L. P. [1983] "On structurally unstable attracting sets of Lorenz attractor type," Trans. Moscow Math. Soc. 44, 153-216.
3. Afraimovich, V. S. [1984] "Strange attractors and quaiattractors," Nonlinear and Turbulent Processes in Physics, 3. ed. Sagdeev, R. Z. (Gordon and Breach. Harwood Academic Publishers), pp. 1133-1138.
4. Afraimovich, V. & Hsu, S.-B. [2002] Lecture on Chaotic Dynamical Systems, AMS/IP, Studies in Advanced Mathematics, Vol. 28.
5. de Melo, W. & van Strien, S. [1993] One-Dimensional Dynamics (Springer-Verlag, NY).
6. Devaney, R. & Nitecki, Z. [1979] "Shift automorphisms in the Hénon mapping," Coram, Math. Phys. 67, 137-146.
7. Gardini, L., Abraham, R., Record, R. J. & Fournier-Prunaret, D. [1994] "A double logistic map." Int. J. Bifurcation and Chaos 4, 145-176.
8. Gavrilov, N. K. & Shilnikov, L. P. [1972] "On threedimensional dynamical systems close to systems with a structurally unstable homoclinic curve, Part 1," Math. USSR Sb. 17, 467-485.
9. Gavrilov, N. K. & Shilnikov, L. P. [1973] "On threedimensional dynamical systems close to systems with a structurally unstable homoclinic curve. Part 2," Math. USSR Sb. 19, 139-156.
10. Gonchenko, S. V., Shilnikov, L. P. & Turaev, D. V. [1993] "Dynamical phenomena in systems with structurally unstable Poincare homoclinic orbits," Russian Acad. Sci. Dokl. Math. 47, 410-415.
11. Gonchenko, S. V., Shilnikov, L. P. & Turaev, D. V. [1996] "Dynamical phenomena in systems with structurally unstable Poincare homoclinic orbits," Chaos 6, 15-31.
12. Gonchenko, S. V. & Gonchenko, V. S. [2000] "On Andronov-Hopf bifurcations of two-dimensional diffeomorphisms with homoclinic tangencies," Weierstrass Institute for Applied Analysis and Stochastics-Preprint No. 556, Berlin, .
13. Gonchenko, S. V. & Gonchenko, V. S. [2004] "On bifurcations of birth of closed invariant curves in the case of two-dimensional diffeomorphisms with homoclinic tangencies," Proc. Math. Steklov Inst. 244, 80-105.
14. Gonchenko, V. S., Kuznetsov, Yu. A. & Meijer, H. G. E. [2005a] "Generalized Henon map and bifurcations of homoclinic tangencies," SIAM J. Appl. Dyn. Sys. 4, 407-436.
15. Gonchenko, S. V., Shilnikov, L. P. & Turaev, D. V. [2005b] "On the dynamical properties of diffeomorphisms with homoclinic tangencies," J. Math. Sci. (N. Y.) 126, 1317-1343.
16. Grines, V. Z. [1975] "On topological conjugacy of diffeomorphisms of a two-manifold on one-dimensional basic sets. Part I," Trans. Moscow Math. Soc. 32, 35-60.
17. Hénon, M. [1976] "A two-dimensional mapping with a strange attractor," Coram. Math. Phys. 50, 69-77.
18. Juang, J., Li, M.-C. & Malkin, M. [2005] "Multidimensional perturbations of chaotic one-dimensional maps and their applications," Preprint of Institute of Mathematics, Academia Sinica, MIAS-2005, Taipei.
19. Li, M.-C. & Malkin, M. [2004] "Bounded nonwandering sets for polynomial mappings," J. Dyn. Contr. Sys. 10, 377-389.
20. Li, M.-C. & Malkin, M. [2006] "Topological horseshoes for perturbations of singular difference equations," Nonlinearity 19, 795-811.
21. n," J. Math. Anal. Appl. 63, 199-223.
22. Misiurewicz, M. & Zgliczynski, P. [2001] "Topological entropy for multidimensional perturbations of onedimensional maps," Int. J. Bifurcation and Chaos 11, 1443-1446.
23. Mira, C. [1987] Chaotic Dynamics (World Scientific, Singapore).
24. Mira, C. Gardini, L., Barugola, A. & Cathala, J.-C. [1996] Chaotic Dynamics in Two-Dimensional Noninvertible Maps (World Scientific, Singapore).
25. Palis, J. & Takens, F. [1993] Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations (Cambridge Univ. Press).
26. Robinson, C. [1999] Dynamical Systems: Stability, Symbolic Dynamics and Chaos, 2nd edition (CRC Press, Boca Raton).
27. Shilnikov, L. P., Shilnikov, A. L., Turaev, D. V. & Chua, L. O. [1998] Methods of Qualitative Theory in Nonlinear Dynamics, Part I (World Scientific, Singapore).
28. Shilnikov, L. P., Shilnikov, A. L., Turaev, D. V. & Chua, L. O. [2001] Methods of Qualitative Theory in Nonlinear Dynamics, Part II (World Scientific, Singapore).
29. Tedeschini-Lalli, L. & Yorke, J. A. [1986] "How often do simple dynamical processes have infinitely many coexisting sinks?" Commun. Math. Phys. 106, 635-657.