October  2019, 24(10): 5597-5600. doi: 10.3934/dcdsb.2019073

Minimal forward random point attractors need not exist

Technische Universität Berlin, Fak. Ⅱ, Institut für Mathematik, Sekr. MA 7-5, Straẞe des 17. Juni 136, Germany

Received  August 2018 Revised  November 2018 Published  April 2019

It is well-known that random attractors of a random dynamical system are generally not unique. It was shown in [1] that if there exist more than one pullback or weak random attractor which attracts a given family of (possibly random) sets, then there exists a minimal (in the sense of smallest) one. This statement does not hold for forward random attractors. The same paper contains an example of a random dynamical system and a deterministic family of sets which has more than one forward attractor which attracts the given family but no minimal one. The question whether one can find an example which has multiple forward point attractors but no minimal one remained open. Here we provide such an example.

Citation: Michael Scheutzow. Minimal forward random point attractors need not exist. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5597-5600. doi: 10.3934/dcdsb.2019073
References:
[1]

H. Crauel and M. Scheutzow, Minimal random attractors, J. Differential Equations, 265 (2018), 702-718.  doi: 10.1016/j.jde.2018.03.011.  Google Scholar

show all references

References:
[1]

H. Crauel and M. Scheutzow, Minimal random attractors, J. Differential Equations, 265 (2018), 702-718.  doi: 10.1016/j.jde.2018.03.011.  Google Scholar

[1]

Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021031

[2]

Jiacheng Wang, Peng-Fei Yao. On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021043

[3]

Brahim Alouini. Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021013

[4]

Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1059-1076. doi: 10.3934/cpaa.2021006

[5]

Scott Schmieding, Rodrigo Treviño. Random substitution tilings and deviation phenomena. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3869-3902. doi: 10.3934/dcds.2021020

[6]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311

[7]

Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034

[8]

Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2891-2905. doi: 10.3934/dcds.2020390

[9]

Yuta Tanoue. Improved Hoeffding inequality for dependent bounded or sub-Gaussian random variables. Probability, Uncertainty and Quantitative Risk, 2021, 6 (1) : 53-60. doi: 10.3934/puqr.2021003

[10]

Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055

[11]

Lingyu Li, Zhang Chen. Asymptotic behavior of non-autonomous random Ginzburg-Landau equation driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3303-3333. doi: 10.3934/dcdsb.2020233

[12]

Xuping Zhang. Pullback random attractors for fractional stochastic $ p $-Laplacian equation with delay and multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021107

[13]

Xu Zhang, Xiang Li. Modeling and identification of dynamical system with Genetic Regulation in batch fermentation of glycerol. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 393-403. doi: 10.3934/naco.2015.5.393

[14]

Francis Hounkpe, Gregory Seregin. An approximation of forward self-similar solutions to the 3D Navier-Stokes system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021059

[15]

Quan Hai, Shutang Liu. Mean-square delay-distribution-dependent exponential synchronization of chaotic neural networks with mixed random time-varying delays and restricted disturbances. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3097-3118. doi: 10.3934/dcdsb.2020221

[16]

Qiao Liu. Partial regularity and the Minkowski dimension of singular points for suitable weak solutions to the 3D simplified Ericksen–Leslie system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021041

[17]

Mauricio Achigar. Extensions of expansive dynamical systems. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3093-3108. doi: 10.3934/dcds.2020399

[18]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2739-2776. doi: 10.3934/dcds.2020384

[19]

Andrés Contreras, Juan Peypouquet. Forward-backward approximation of nonlinear semigroups in finite and infinite horizon. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021051

[20]

Ugo Bessi. Another point of view on Kusuoka's measure. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3241-3271. doi: 10.3934/dcds.2020404

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (114)
  • HTML views (334)
  • Cited by (0)

Other articles
by authors

[Back to Top]