American Institute of Mathematical Sciences

Advanced
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 & 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).   Google Scholar

[2]

A. M. Ateiwi, About bounded solutions of linear stochastic Ito systems,, Miskolc Math. Notes, 3 (2002), 3.   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.  doi: 10.1142/S0219493702000364.  Google Scholar

[5]

R. Khasminskii, Stochastic Stability of Differential Equations,, Second edition, (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.  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.  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, (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, (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.  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.  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.  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.  doi: 10.1007/s10884-013-9340-3.  Google Scholar

show all references

References:
[1]

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

[2]

A. M. Ateiwi, About bounded solutions of linear stochastic Ito systems,, Miskolc Math. Notes, 3 (2002), 3.   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.  doi: 10.1142/S0219493702000364.  Google Scholar

[5]

R. Khasminskii, Stochastic Stability of Differential Equations,, Second edition, (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.  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.  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, (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, (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.  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.  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.  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.  doi: 10.1007/s10884-013-9340-3.  Google Scholar

[1]

Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324
[2]

Jie Li, Xiangdong Ye, Tao Yu. Mean equicontinuity, complexity and applications. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 359-393. doi: 10.3934/dcds.2020167
[3]

Timothy Chumley, Renato Feres. Entropy production in random billiards. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1319-1346. doi: 10.3934/dcds.2020319
[4]

Kai Yang. Scattering of the focusing energy-critical NLS with inverse square potential in the radial case. Communications on Pure & Applied Analysis, 2021, 20 (1) : 77-99. doi: 10.3934/cpaa.2020258
[5]

Patrick W. Dondl, Martin Jesenko. Threshold phenomenon for homogenized fronts in random elastic media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 353-372. doi: 10.3934/dcdss.2020329
[6]

Illés Horváth, Kristóf Attila Horváth, Péter Kovács, Miklós Telek. Mean-field analysis of a scaling MAC radio protocol. Journal of Industrial & Management Optimization, 2021, 17 (1) : 279-297. doi: 10.3934/jimo.2019111
[7]

Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106
[8]

Laura Aquilanti, Simone Cacace, Fabio Camilli, Raul De Maio. A Mean Field Games model for finite mixtures of Bernoulli and categorical distributions. Journal of Dynamics & Games, 2020  doi: 10.3934/jdg.2020033
[9]

Petr Pauš, Shigetoshi Yazaki. Segmentation of color images using mean curvature flow and parametric curves. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1123-1132. doi: 10.3934/dcdss.2020389
[10]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121
[11]

Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020390
[12]

Josselin Garnier, Knut Sølna. Enhanced Backscattering of a partially coherent field from an anisotropic random lossy medium. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1171-1195. doi: 10.3934/dcdsb.2020158
[13]

Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3093-3116. doi: 10.3934/dcds.2020039
[14]

Alain Bensoussan, Xinwei Feng, Jianhui Huang. Linear-quadratic-Gaussian mean-field-game with partial observation and common noise. Mathematical Control & Related Fields, 2021, 11 (1) : 23-46. doi: 10.3934/mcrf.2020025
[15]

Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026
[16]

Jiannan Zhang, Ping Chen, Zhuo Jin, Shuanming Li. Open-loop equilibrium strategy for mean-variance portfolio selection: A log-return model. Journal of Industrial & Management Optimization, 2021, 17 (2) : 765-777. doi: 10.3934/jimo.2019133
[17]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080
[18]

Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168
[19]

Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352
[20]

Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020284

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (60)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences