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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

Thai Son Doan 1, , Martin Rasmussen 1, and  Peter E. Kloeden 2,

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.
Keywords: mean-square random dynamical system, bifurcation., mean-square random attractor, Dichotomy spectrum.
Mathematics Subject Classification: Primary: 37H15, 37H20, 60H10; Secondary: 37H10, 60H3.
Citation: Thai Son Doan, Martin Rasmussen, Peter E. Kloeden. The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 875-887. doi: 10.3934/dcdsb.2015.20.875
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.

show all references

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.

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