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On Lyapunov exponents of difference equations with random delay
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 |
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer, Berlin, Heidelberg, New York, 1998. |
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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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