May  2015, 20(3): 875-887. doi: 10.3934/dcdsb.2015.20.875

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, United Kingdom, United Kingdom

2. 

School of Mathematics and Statistics, Huazhong University of Science & Technology, Wuhan 430074, China

Received  November 2013 Revised  April 2014 Published  January 2015

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.
Citation: Thai Son Doan, Martin Rasmussen, Peter E. Kloeden. The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 875-887. doi: 10.3934/dcdsb.2015.20.875
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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.  Google Scholar

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show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer, Berlin, Heidelberg, New York, 1998. Google Scholar

[2]

A. M. Ateiwi, About bounded solutions of linear stochastic Ito systems, Miskolc Math. Notes, 3 (2002), 3-12.  Google Scholar

[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., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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