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
References:
[1]

L. Arnold, Random Dynamical Systems,, Springer, (1998).

[2]

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

[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. doi: 10.1142/S0219493702000364.

[5]

R. Khasminskii, Stochastic Stability of Differential Equations,, Second edition, (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. 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. doi: 10.1016/j.jde.2012.05.016.

[8]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations,, Applications of Mathematics, (1992). doi: 10.1007/978-3-662-12616-5.

[9]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems,, Mathematical Surveys and Monographs, (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. 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. 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. 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. doi: 10.1007/s10884-013-9340-3.

show all references

References:
[1]

L. Arnold, Random Dynamical Systems,, Springer, (1998).

[2]

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

[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. doi: 10.1142/S0219493702000364.

[5]

R. Khasminskii, Stochastic Stability of Differential Equations,, Second edition, (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. 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. doi: 10.1016/j.jde.2012.05.016.

[8]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations,, Applications of Mathematics, (1992). doi: 10.1007/978-3-662-12616-5.

[9]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems,, Mathematical Surveys and Monographs, (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. 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. 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. 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. doi: 10.1007/s10884-013-9340-3.

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