August  2015, 35(8): 3745-3769. doi: 10.3934/dcds.2015.35.3745

Stochastic bifurcation of pathwise random almost periodic and almost automorphic solutions for random dynamical systems

1. 

Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801

Received  May 2014 Revised  December 2014 Published  February 2015

In this paper, we introduce concepts of pathwise random almost periodic and almost automorphic solutions for dynamical systems generated by non-autonomous stochastic equations. These solutions are pathwise stochastic analogues of deterministic dynamical systems. The existence and bifurcation of random periodic (random almost periodic, random almost automorphic) solutions have been established for a one-dimensional stochastic equation with multiplicative noise.
Citation: Bixiang Wang. Stochastic bifurcation of pathwise random almost periodic and almost automorphic solutions for random dynamical systems. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3745-3769. doi: 10.3934/dcds.2015.35.3745
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show all references

References:
[1]

Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

Diffusion Processes and Related Problems in Analysis, Vol II: Stochastic Flows, Birkhauser, Boston, 27 (1992), 241-255.  Google Scholar

[3]

Advances in Nonlinear Stochastic Mechanics, Kluwer Acad. Publ., Dordrecht, 47 (1996), 19-28. doi: 10.1007/978-94-009-0321-0_3.  Google Scholar

[4]

North-Holland, Amsterdam, 1992.  Google Scholar

[5]

Stoch. Dyn., 6 (2006), 1-21. doi: 10.1142/S0219493706001621.  Google Scholar

[6]

J. Differential Equations, 246 (2009), 845-869. doi: 10.1016/j.jde.2008.05.017.  Google Scholar

[7]

Discrete Continuous Dynamical Systems, 21 (2008), 415-443. doi: 10.3934/dcds.2008.21.415.  Google Scholar

[8]

Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455. doi: 10.3934/dcdsb.2010.14.439.  Google Scholar

[9]

Nonlinear Anal., 74 (2011), 3671-3684. doi: 10.1016/j.na.2011.02.047.  Google Scholar

[10]

Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 10 (2003), 491-513.  Google Scholar

[11]

Set-Valued Analysis, 11 (2003), 153-201. doi: 10.1023/A:1022902802385.  Google Scholar

[12]

Lecture Notes in Mathematics, 1779, Springer, Berlin, 2002. doi: 10.1007/b83277.  Google Scholar

[13]

Probab. Th. Re. Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.  Google Scholar

[14]

J. Dyn. Diff. Eqns., 10 (1998), 259-274. doi: 10.1023/A:1022665916629.  Google Scholar

[15]

Comm. Math. Sci., 1 (2003), 133-151. doi: 10.4310/CMS.2003.v1.n1.a9.  Google Scholar

[16]

Lecture Notes in Mathematics 377, Springer-Verlag, New York, 1974.  Google Scholar

[17]

Stoch. Stoch. Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083.  Google Scholar

[18]

J. Dynam. Differential Equations, 23 (2011), 671-681. doi: 10.1007/s10884-011-9222-5.  Google Scholar

[19]

American Mathematical Society, Providence, RI, 1988.  Google Scholar

[20]

Discrete and Continuous Dynamical Systems, 24 (2009), 855-882. doi: 10.3934/dcds.2009.24.855.  Google Scholar

[21]

Cambridge University Press, Cambridge, 1982.  Google Scholar

[22]

Transactions of American Mathematical Society, 364 (2012), 3781-3804. doi: 10.1090/S0002-9947-2012-05555-3.  Google Scholar

[23]

in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, Dresden, (1992), 185-192. Google Scholar

[24]

Van Nostrand Reinhold, London, 1971.  Google Scholar

[25]

Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[26]

Mem. Amer. Math. Soc., 136 (1998), 1-93. doi: 10.1090/memo/0647.  Google Scholar

[27]

J. Differential Equations, 122 (1995), 114-136. doi: 10.1006/jdeq.1995.1141.  Google Scholar

[28]

J. Differential Equations, 122 (1995), 373-397. doi: 10.1006/jdeq.1995.1152.  Google Scholar

[29]

Trans. Amer. Math. Soc., 347 (1995), 4413-4431. doi: 10.1090/S0002-9947-1995-1311916-9.  Google Scholar

[30]

J. Dynamics and Differential Equations, 8 (1996), 299-323. doi: 10.1007/BF02218894.  Google Scholar

[31]

Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[32]

J. Differential Equations, 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015.  Google Scholar

[33]

Discrete and Continuous Dynamical Systems Series A, 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269.  Google Scholar

[34]

Nonlinear Analysis TMA, 103 (2014), 9-25. doi: 10.1016/j.na.2014.02.013.  Google Scholar

[35]

Nonlinearity, 22 (2009), 765-782. doi: 10.1088/0951-7715/22/4/005.  Google Scholar

[36]

Rocky Mountain Journal of Mathematics, 18 (1988), 479-494. doi: 10.1216/RMJ-1988-18-2-479.  Google Scholar

[37]

Random and Computational Dynamics, 1 (1993), 277-305.  Google Scholar

[38]

Springer-Verlag, New York, 1975.  Google Scholar

[39]

Pitman Research Notes in Mathematics Series 321, Longman Group Limited, England, 1995.  Google Scholar

[40]

J. Differential Equations, 246 (2009), 2020-2038. doi: 10.1016/j.jde.2008.10.011.  Google Scholar

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