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A sufficient condition for bifurcation in random dynamical systems
Author(s):
Xiaopeng
Chen;
Jinqiao
Duan;
Xinchu
Fu
Journal:
Proc. Amer. Math. Soc.
138
(2010),
965-973.
MSC (2000):
Primary 37H20, 37B30, 60H10
Posted:
October 19, 2009
MathSciNet review:
2566563
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Abstract:
Some properties of the random Conley index are obtained, and then a sufficient condition for the existence of abstract bifurcation points for both discrete-time and continuous-time random dynamical systems is presented. This stochastic bifurcation phenomenon is demonstrated by a few examples.
References:
-
- 1.
- L. Arnold, Random Dynamical Systems, Springer, Berlin-Heidelberg-New York, 1998. MR 1723992 (2000m:37087)
- 2.
- T. Bartsch, The Conley Index over a Space, Mathematische Zeitschrift 209(1992), 167-177. MR 1147812 (92m:58019)
- 3.
- I. Chueshov, Monotone Dynamical Systems Theory and Applications, Springer Lecture Notes in Mathematics, vol. 1779, Berlin-Heidelberg-New York, 2002. MR 1902500 (2003d:37072)
- 4.
- C. Conley, Isolated Invariant Sets and the Morse Index, Conf. Board Math. Sci., vol. 38, Amer. Math. Soc., Providence, RI, 1978. MR 511133 (80c:58009)
- 5.
- J. Franks and D. Richeson, Shift Equivalence and the Conley Index, Transactions of the American Mathematical Society 352(2000), 3305-3322. MR 1665329 (2000j:37013)
- 6.
- X. C. Fu and K. H. Xu, The Conley Index and Bifurcation Points, Nonlinear Analysis: Theory, Methods & Applications 19(1992), 1137-1142. MR 1195047 (93j:58096)
- 7.
- M. Izydorek and S. Rybicki, Bifurcations of Bounded Solutions of
 -Parameter ODE's, Journal of Differential Equations 130(1996), 267-276. MR 1410887 (97i:34053) - 8.
- E. Kappos, The Conley Index and Global Bifurcations, Part I: Concepts and Theory, International Journal of Bifurcation and Chaos 5(1995), 937-953. MR 1348294 (96f:58112)
- 9.
- E. Kappos, The Conley Index and Global Bifurcations. II. Illustrative Applications, International Journal of Bifurcation and Chaos 6(1996), 2491-2505. MR 1450272 (98d:58129)
- 10.
- P. Krzysztof and Rybakowski, The Homotopy Index and Partial Differential Equations, Springer, Berlin-Heidelberg-New York, 1987. MR 910097 (89d:58025)
- 11.
- M. Kunze, Non-Smooth Dynamical Systems, Springer Lecture Notes in Mathematics, vol. 1744, Berlin-Heidelberg-New York, 2000. MR 1789550 (2002e:34002)
- 12.
- Z. X. Liu, Conley Index for Random Dynamical Systems, Journal of Differential Equations 244(2008), 1603-1628. MR 2404433 (2009g:37048)
- 13.
- K. Mischaikow and M. Mrozek, Conley Index. Handbook of dynamical systems, vol. 2, 393-460, North-Holland, Amsterdam, 2002. MR 1901060 (2003g:37022)
- 14.
- J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, Berlin-Heidelberg-New York, 1983. MR 688146 (84d:35002)
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Additional Information:
Xiaopeng
Chen
Affiliation:
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
Email:
chenxiao002214336@yahoo.cn
Jinqiao
Duan
Affiliation:
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Illinois 60616
Email:
duan@iit.edu
Xinchu
Fu
Affiliation:
Department of Mathematics, Shanghai University, Shanghai 200444, People's Republic of China
Email:
xcfu@shu.edu.cn
DOI:
10.1090/S0002-9939-09-10093-X
PII:
S 0002-9939(09)10093-X
Keywords:
Random dynamical systems,
discrete-time and continuous-time dynamical systems,
random homeomorphism,
Conley index,
abstract bifurcation point
Received by editor(s):
April 19, 2009,
Received by editor(s) in revised form:
June 24, 2009
Posted:
October 19, 2009
Additional Notes:
The first author would like to thank Zhenxin Liu for helpful comments.
The second author was supported in part by NSF Grant 0620539, the Cheung Kong Scholars Program and the K. C. Wong Education Foundation.
The third author was supported in part by NSFC Grant 10672146.
Communicated by:
Yingfei Yi
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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