# American Institute of Mathematical Sciences

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 - A, 2015, 35 (8) : 3745-3769. doi: 10.3934/dcds.2015.35.3745
##### References:
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Differential Equations, 246 (2009), 845.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar [7] T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness,, Discrete Continuous Dynamical Systems, 21 (2008), 415.  doi: 10.3934/dcds.2008.21.415.  Google Scholar [8] T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439.  doi: 10.3934/dcdsb.2010.14.439.  Google Scholar [9] T. Caraballo, M. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion,, Nonlinear Anal., 74 (2011), 3671.  doi: 10.1016/j.na.2011.02.047.  Google Scholar [10] T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems,, Dynamics of Continuous, 10 (2003), 491.   Google Scholar [11] T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems,, Set-Valued Analysis, 11 (2003), 153.  doi: 10.1023/A:1022902802385.  Google Scholar [12] I. Chueshow, Monotone Random Systems-Theory and Applications,, Lecture Notes in Mathematics, 1779 (2002).  doi: 10.1007/b83277.  Google Scholar [13] H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Th. Re. Fields, 100 (1994), 365.  doi: 10.1007/BF01193705.  Google Scholar [14] H. Crauel and F. Flandoli, Additive noise destroys a pitchfork bifurcation,, J. Dyn. Diff. Eqns., 10 (1998), 259.  doi: 10.1023/A:1022665916629.  Google Scholar [15] J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary,, Comm. Math. Sci., 1 (2003), 133.  doi: 10.4310/CMS.2003.v1.n1.a9.  Google Scholar [16] A. M. Fink, Almost Periodic Differential Equations,, Lecture Notes in Mathematics 377, (1974).   Google Scholar [17] F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise,, Stoch. Stoch. Rep., 59 (1996), 21.  doi: 10.1080/17442509608834083.  Google Scholar [18] M. J. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion,, J. Dynam. Differential Equations, 23 (2011), 671.  doi: 10.1007/s10884-011-9222-5.  Google Scholar [19] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, American Mathematical Society, (1988).   Google Scholar [20] J. Huang and W. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains,, Discrete and Continuous Dynamical Systems, 24 (2009), 855.  doi: 10.3934/dcds.2009.24.855.  Google Scholar [21] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations,, Cambridge University Press, (1982).   Google Scholar [22] Q. Liu and Y. Wang, Phase-translation group actions on strongly monotone skew-product semiflows,, Transactions of American Mathematical Society, 364 (2012), 3781.  doi: 10.1090/S0002-9947-2012-05555-3.  Google Scholar [23] B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations,, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, (1992), 185.   Google Scholar [24] G. R. Sell, Topological Dynamics and Ordinary Differential Equations,, Van Nostrand Reinhold, (1971).   Google Scholar [25] R. Sell and Y. You, Dynamics of Evolutionary Equations,, Springer-Verlag, (2002).  doi: 10.1007/978-1-4757-5037-9.  Google Scholar [26] W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows,, Mem. Amer. Math. Soc., 136 (1998), 1.  doi: 10.1090/memo/0647.  Google Scholar [27] W. Shen and Y. Yi, Dynamics of almost periodic scalar parabolic equations,, J. Differential Equations, 122 (1995), 114.  doi: 10.1006/jdeq.1995.1141.  Google Scholar [28] W. Shen and Y. Yi, Asymptotic almost periodicity of scalar parabolic equations with almost periodic time dependence,, J. Differential Equations, 122 (1995), 373.  doi: 10.1006/jdeq.1995.1152.  Google Scholar [29] W. Shen and Y. Yi, On minimal sets of scalar parabolic equations with skew-product structures,, Trans. Amer. Math. Soc., 347 (1995), 4413.  doi: 10.1090/S0002-9947-1995-1311916-9.  Google Scholar [30] W. Shen and Y. Yi, Ergodicity of minimal sets in scalar parabolic equations,, J. Dynamics and Differential Equations, 8 (1996), 299.  doi: 10.1007/BF02218894.  Google Scholar [31] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997).  doi: 10.1007/978-1-4612-0645-3.  Google Scholar [32] B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems,, J. Differential Equations, 253 (2012), 1544.  doi: 10.1016/j.jde.2012.05.015.  Google Scholar [33] B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise,, Discrete and Continuous Dynamical Systems Series A, 34 (2014), 269.  doi: 10.3934/dcds.2014.34.269.  Google Scholar [34] B. Wang, Existence, stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations,, Nonlinear Analysis TMA, 103 (2014), 9.  doi: 10.1016/j.na.2014.02.013.  Google Scholar [35] Y. Wang, Asymptotic symmetry in strongly monotone skew-product semiflows with applications,, Nonlinearity, 22 (2009), 765.  doi: 10.1088/0951-7715/22/4/005.  Google Scholar [36] J. R. Ward Jr., Bounded and almost periodic solutions of semi-linear parabolic equations,, Rocky Mountain Journal of Mathematics, 18 (1988), 479.  doi: 10.1216/RMJ-1988-18-2-479.  Google Scholar [37] K. Xu, Bifurcations of random differential equations in dimension one,, Random and Computational Dynamics, 1 (1993), 277.   Google Scholar [38] T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions,, Springer-Verlag, (1975).   Google Scholar [39] S. Zaidman, Topics in Abstract Differential Equations II,, Pitman Research Notes in Mathematics Series 321, (1995).   Google Scholar [40] H. Zhao and Z. Zheng, Random periodic solutions of random dynamical systems,, J. Differential Equations, 246 (2009), 2020.  doi: 10.1016/j.jde.2008.10.011.  Google Scholar

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##### References:
 [1] L. Arnold, Random Dynamical Systems,, Springer Monographs in Mathematics, (1998).  doi: 10.1007/978-3-662-12878-7.  Google Scholar [2] L. Arnold and P. Boxler, Stochastic bifurcation: instructive examples in dimension one,, Diffusion Processes and Related Problems in Analysis, 27 (1992), 241.   Google Scholar [3] L. Arnold and B. Schmalfuss, Fixed points and attractors for random dynamical systems,, Advances in Nonlinear Stochastic Mechanics, 47 (1996), 19.  doi: 10.1007/978-94-009-0321-0_3.  Google Scholar [4] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland, (1992).   Google Scholar [5] P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems,, Stoch. Dyn., 6 (2006), 1.  doi: 10.1142/S0219493706001621.  Google Scholar [6] P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains,, J. Differential Equations, 246 (2009), 845.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar [7] T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness,, Discrete Continuous Dynamical Systems, 21 (2008), 415.  doi: 10.3934/dcds.2008.21.415.  Google Scholar [8] T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439.  doi: 10.3934/dcdsb.2010.14.439.  Google Scholar [9] T. Caraballo, M. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion,, Nonlinear Anal., 74 (2011), 3671.  doi: 10.1016/j.na.2011.02.047.  Google Scholar [10] T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems,, Dynamics of Continuous, 10 (2003), 491.   Google Scholar [11] T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems,, Set-Valued Analysis, 11 (2003), 153.  doi: 10.1023/A:1022902802385.  Google Scholar [12] I. Chueshow, Monotone Random Systems-Theory and Applications,, Lecture Notes in Mathematics, 1779 (2002).  doi: 10.1007/b83277.  Google Scholar [13] H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Th. Re. Fields, 100 (1994), 365.  doi: 10.1007/BF01193705.  Google Scholar [14] H. Crauel and F. Flandoli, Additive noise destroys a pitchfork bifurcation,, J. Dyn. Diff. Eqns., 10 (1998), 259.  doi: 10.1023/A:1022665916629.  Google Scholar [15] J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary,, Comm. Math. Sci., 1 (2003), 133.  doi: 10.4310/CMS.2003.v1.n1.a9.  Google Scholar [16] A. M. Fink, Almost Periodic Differential Equations,, Lecture Notes in Mathematics 377, (1974).   Google Scholar [17] F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise,, Stoch. Stoch. Rep., 59 (1996), 21.  doi: 10.1080/17442509608834083.  Google Scholar [18] M. J. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion,, J. Dynam. Differential Equations, 23 (2011), 671.  doi: 10.1007/s10884-011-9222-5.  Google Scholar [19] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, American Mathematical Society, (1988).   Google Scholar [20] J. Huang and W. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains,, Discrete and Continuous Dynamical Systems, 24 (2009), 855.  doi: 10.3934/dcds.2009.24.855.  Google Scholar [21] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations,, Cambridge University Press, (1982).   Google Scholar [22] Q. Liu and Y. Wang, Phase-translation group actions on strongly monotone skew-product semiflows,, Transactions of American Mathematical Society, 364 (2012), 3781.  doi: 10.1090/S0002-9947-2012-05555-3.  Google Scholar [23] B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations,, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, (1992), 185.   Google Scholar [24] G. R. Sell, Topological Dynamics and Ordinary Differential Equations,, Van Nostrand Reinhold, (1971).   Google Scholar [25] R. Sell and Y. You, Dynamics of Evolutionary Equations,, Springer-Verlag, (2002).  doi: 10.1007/978-1-4757-5037-9.  Google Scholar [26] W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows,, Mem. Amer. Math. Soc., 136 (1998), 1.  doi: 10.1090/memo/0647.  Google Scholar [27] W. Shen and Y. Yi, Dynamics of almost periodic scalar parabolic equations,, J. Differential Equations, 122 (1995), 114.  doi: 10.1006/jdeq.1995.1141.  Google Scholar [28] W. Shen and Y. Yi, Asymptotic almost periodicity of scalar parabolic equations with almost periodic time dependence,, J. Differential Equations, 122 (1995), 373.  doi: 10.1006/jdeq.1995.1152.  Google Scholar [29] W. Shen and Y. Yi, On minimal sets of scalar parabolic equations with skew-product structures,, Trans. Amer. Math. Soc., 347 (1995), 4413.  doi: 10.1090/S0002-9947-1995-1311916-9.  Google Scholar [30] W. Shen and Y. Yi, Ergodicity of minimal sets in scalar parabolic equations,, J. Dynamics and Differential Equations, 8 (1996), 299.  doi: 10.1007/BF02218894.  Google Scholar [31] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997).  doi: 10.1007/978-1-4612-0645-3.  Google Scholar [32] B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems,, J. Differential Equations, 253 (2012), 1544.  doi: 10.1016/j.jde.2012.05.015.  Google Scholar [33] B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise,, Discrete and Continuous Dynamical Systems Series A, 34 (2014), 269.  doi: 10.3934/dcds.2014.34.269.  Google Scholar [34] B. Wang, Existence, stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations,, Nonlinear Analysis TMA, 103 (2014), 9.  doi: 10.1016/j.na.2014.02.013.  Google Scholar [35] Y. Wang, Asymptotic symmetry in strongly monotone skew-product semiflows with applications,, Nonlinearity, 22 (2009), 765.  doi: 10.1088/0951-7715/22/4/005.  Google Scholar [36] J. R. Ward Jr., Bounded and almost periodic solutions of semi-linear parabolic equations,, Rocky Mountain Journal of Mathematics, 18 (1988), 479.  doi: 10.1216/RMJ-1988-18-2-479.  Google Scholar [37] K. Xu, Bifurcations of random differential equations in dimension one,, Random and Computational Dynamics, 1 (1993), 277.   Google Scholar [38] T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions,, Springer-Verlag, (1975).   Google Scholar [39] S. Zaidman, Topics in Abstract Differential Equations II,, Pitman Research Notes in Mathematics Series 321, (1995).   Google Scholar [40] H. Zhao and Z. Zheng, Random periodic solutions of random dynamical systems,, J. Differential Equations, 246 (2009), 2020.  doi: 10.1016/j.jde.2008.10.011.  Google Scholar
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