The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor

Thai Son Doan; Martin Rasmussen; Peter E. Kloeden

doi:10.3934/dcdsb.2015.20.875

Article Contents

2015, Volume 20, Issue 3: 875-887. Doi: 10.3934/dcdsb.2015.20.875

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The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor

1.
Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ
2.
School of Mathematics and Statistics, Huazhong University of Science & Technology, Wuhan 430074

Received: November 2013

Revised: April 2014

Published: May 2015

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Abstract

The dichotomy spectrum is introduced for linear mean-square random dynamical systems, and it is shown that for finite-dimensional mean-field stochastic differential equations, the dichotomy spectrum consists of finitely many compact intervals. It is then demonstrated that a change in the sign of the dichotomy spectrum is associated with a bifurcation from a trivial to a non-trivial mean-square random attractor.

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Mathematics Subject Classification: Primary: 37H15, 37H20, 60H10; Secondary: 37H10, 60H30.

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References

[1]	L. Arnold, Random Dynamical Systems, Springer, Berlin, Heidelberg, New York, 1998.
[2]	A. M. Ateiwi, About bounded solutions of linear stochastic Ito systems, Miskolc Math. Notes, 3 (2002), 3-12.
[3]	M. Callaway, T. S. Doan, J. S. W. Lamb and M. Rasmussen, The dichotomy spectrum for random dynamical systems and pitchfork bifurcations with bounded noise, submitted.
[4]	N. D. Cong and S. Siegmund, Dichotomy spectrum of nonautonomous linear stochastic differential equations, Stochastics and Dynamics, 2 (2002), 175-201.doi: 10.1142/S0219493702000364.
[5]	R. Khasminskii, Stochastic Stability of Differential Equations, Second edition, Stochastic Modelling and Applied Probability, 66, Springer, Heidelberg, 2012.doi: 10.1007/978-3-642-23280-0.
[6]	P. E. Kloeden and T. Lorenz, Stochastic differential equations with nonlocal sample dependence, Stochastic Analysis and Applications, 28 (2010), 937-945.doi: 10.1080/07362994.2010.515194.
[7]	P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, Journal of Differential Equations, 253 (2012), 1422-1438.doi: 10.1016/j.jde.2012.05.016.
[8]	P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics, 23, Springer, Berlin, 1992.doi: 10.1007/978-3-662-12616-5.
[9]	P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176, American Mathematical Society, Providence, RI, 2011.doi: 10.1090/surv/176.
[10]	P. E. Kloeden and P. Marín-Rubio, Negatively invariant sets and entire trajectories of set-valued dynamical systems, Set-Valued and Variational Analysis, 19 (2011), 43-57.doi: 10.1007/s11228-009-0123-2.
[11]	R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, Journal of Differential Equations, 27 (1978), 320-358.doi: 10.1016/0022-0396(78)90057-8.
[12]	D. Stoica, Uniform exponential dichotomy of stochastic cocycles, Stochastic Processes and their Applications, 120 (2010), 1920-1928.doi: 10.1016/j.spa.2010.05.016.
[13]	G. Wang and Y. Cao, Dynamical spectrum in random dynamical systems, Journal of Dynamics and Differential Equations, 26 (2014), 1-20.doi: 10.1007/s10884-013-9340-3.