|
One-parameter families of smooth interval maps: Density of hyperbolicity and robust chaos
Author(s):
Sebastian
van Strien
Journal:
Proc. Amer. Math. Soc.
138
(2010),
4443-4446.
MSC (2010):
Primary 37E05, 37Gxx, 37Dxx
Posted:
June 22, 2010
MathSciNet review:
2680068
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
In this paper we will discuss the notion of robust chaos and show that (i) there are natural one-parameter families of interval maps with robust chaos and (ii) hyperbolicity is dense within generic one-parameter families (and so these families are not robustly chaotic).
References:
-
- 1.
- J.M. Aguirregabiria, Robust chaos with variable Lyapunov exponent in smooth one-dimensional maps, Chaos, Solitons and Fractals 42 (2009), no. 4, 2531-2539.
- 2.
- M. Andrecut and M.K. Ali, Example of robust chaos in a smooth map, Europhysics Letters 54 (2001), no. 3, 300-305.
- 3.
- -, Robust chaos in smooth unimodal maps, Physical Review E 64 (2001), no. 2, 25203. MR 1849767 (2002e:37048)
- 4.
- V. I. Arnol
 d, Catastrophe theory, third ed., Springer-Verlag, Berlin, 1992, translated from the Russian by G. S. Wassermann, based on a translation by R. K. Thomas. MR 1178935 (93h:58018) - 5.
- S. Banerjee, J.A. Yorke, and C. Grebogi, Robust chaos, Physical Review Letters 80 (1998), no. 14, 3049-3052.
- 6.
- Welington de Melo and Sebastian van Strien, A structure theorem in one-dimensional dynamics, Ann. of Math. (2) 129 (1989), no. 3, 519-546. MR 90m:58106
- 7.
- -, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 95a:58035
- 8.
- Z. Elhadj and J.C. Sprott, On the robustness of chaos in dynamical systems: Theories and applications, Frontiers of Physics in China 3 (2008), no. 2, 195-204.
- 9.
- O.S. Kozlovski, W. Shen, and S. van Strien, Rigidity for real polynomials, Ann. of Math. (2) 165 (2007), no. 3, 749-841. MR 2335796 (2008m:37063)
- 10.
- -, Density of hyperbolicity in dimension one, Ann. of Math. (2) 166 (2007), no. 1, 145-182. MR 2342693 (2008j:37081)
- 11.
- E. Lett, Example of robust chaos in a smooth map, Europhysics Letters 54 (2001), no. 3, 300-305.
- 12.
- S. Newhouse, J. Palis, and F. Takens, Bifurcations and stability of families of diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 57 (1983), 5-71. MR 699057 (84g:58080)
- 13.
- Lasse Rempe and Sebastian van Strien, Density of hyperbolicity for real transcendental entire functions with real singular values, in preparation, 2010.
- 14.
- Sebastian van Strien, One-dimensional dynamics in the new millennium, Discr. and Cont. Dyn. Syst. A 27 (2010), no. 2, 557-588.
- 15.
- Sebastian van Strien and Edson Vargas, Real bounds, ergodicity and negative Schwarzian for multimodal maps, J. Amer. Math. Soc. 17 (2004), no. 4, 749-782 (electronic). MR 2083467
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical
Society
with
MSC (2010):
37E05, 37Gxx, 37Dxx
Retrieve articles in all Journals with
MSC (2010):
37E05, 37Gxx, 37Dxx
Additional Information:
Sebastian
van Strien
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Email:
strien@maths.warwick.ac.uk
DOI:
10.1090/S0002-9939-2010-10446-X
PII:
S 0002-9939(2010)10446-X
Received by editor(s):
December 3, 2009
Received by editor(s) in revised form:
February 15, 2010
Posted:
June 22, 2010
Communicated by:
Yingfei Yi
Copyright of article:
Copyright
2010,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
