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

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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

Abstract
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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 & 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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