American Journal of Applied Mathematics

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Local Approximate Forward Attractors of Nonautonomous Dynamical Systems

Received: 06 May 2020    Accepted: 24 July 2020    Published: 25 September 2020
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Abstract

Pullback dynamics of nonautonomous dynamical systems has been considerably developed. However, it is still a tough job to study forward dynamics of nonautonomous dynamical systems, since forward attractors were only obtained in some particular cases. In the paper, under some reasonable conditions, it is shown that closing to a local pullback attractor, there is an approximate forward attractor. Specifically, let ϕ be a cocycle semiflow on a Banach space X with driving system θ on a base space P. Suppose that the base space P is compact and ϕ is uniformly asymptotically compact. Let A(∙) be a local pullback attractor with being compact. We prove that every ε-extended neighborhood Aε(∙) of A(∙) will forward attract every bounded set B(∙) that is pullback attracted by A(∙). We then call Aε(∙) an approximate forward attractor of ϕ.

DOI 10.11648/j.ajam.20200805.16
Published in American Journal of Applied Mathematics (Volume 8, Issue 5, October 2020)
Page(s) 278-283
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Nonautonomous Dynamical Systems, Pullback Attractors, Approximate Forward Attractors

References
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[2] D. Cheban, P. Kloeden and B. Schmalfuss, The relation between pullback and global attractors for nonautonomous dynamical systems, Nonlinear Dyn. Syst. Theory 2 (2002), 125-144.
[3] V. V. Chepyzhov and M. I. Vishik, Attractors of Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI, 2002.
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[5] T. Caraballo, J. C. Jara, J. A. Langa and Z.X. Liu, Morse decomposition of attractors for non-autonomous dynamical systems, Advanced Nonlinear Studies, 13 (2), (2013), 309-329.
[6] T. Caraballo, J. A. Langa and R. Obaya, Pullback, forward and chaotic dynamics in 1D non-autonomous linear-dissipative equations, Nonlinearity 30. 1 (2017), 274-299.
[7] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non autonomous Dynamical Systems, Appl. Math. Sci. 182, 2013.
[8] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields 100, (1994) 365-393.
[9] H. Crauel, A. Debussche and F. Flandoli, Random Attractors, J. Dynam. Differential Equations 9 (1997), 307-341.
[10] X. W. Ju, D. S. Li, C. Q. Li and A. L. Qi, Approximate forward attractors of non-autonomous dynamical systems, Chinese Annals of Mathematics, Series B, 2019, 40 (4), 541-554.
[11] P. E. Kloeden, Asymptotic invariance and the discretisation of non-autonomous forward attracting sets, J. Comput. Dynamics, 3 (2016), 179-189.
[12] P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., (2016), 259-268.
[13] H. Y. Cui and P. E. Kloeden, Invariant forward attractors of non-autonomous random dynamical systems, Journal of Differential Equations 265.12 (2018), 6166-6186.
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[15] A. N. Carvalho, J. A. Langa, J. C. Robinson and Suarez, Characterization of nonautonomous attractors of a perturbed gradient system, J. Differential Equations 236 (2007), 570-603.
[16] M. Rasmussen, Morse decompositions of non-autonomous dynamical systems, Trans. Amer. Math. Soc. 359 (2007), no. 10, 5091-5115.
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Author Information
  • Department of Mathematics, Civil Aviation University of China, Tianjin, China

  • Department of Mathematics, Civil Aviation University of China, Tianjin, China

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    Ailing Qi, Xuewei Ju. (2020). Local Approximate Forward Attractors of Nonautonomous Dynamical Systems. American Journal of Applied Mathematics, 8(5), 278-283. https://doi.org/10.11648/j.ajam.20200805.16

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    Ailing Qi; Xuewei Ju. Local Approximate Forward Attractors of Nonautonomous Dynamical Systems. Am. J. Appl. Math. 2020, 8(5), 278-283. doi: 10.11648/j.ajam.20200805.16

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    AMA Style

    Ailing Qi, Xuewei Ju. Local Approximate Forward Attractors of Nonautonomous Dynamical Systems. Am J Appl Math. 2020;8(5):278-283. doi: 10.11648/j.ajam.20200805.16

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  • @article{10.11648/j.ajam.20200805.16,
      author = {Ailing Qi and Xuewei Ju},
      title = {Local Approximate Forward Attractors of Nonautonomous Dynamical Systems},
      journal = {American Journal of Applied Mathematics},
      volume = {8},
      number = {5},
      pages = {278-283},
      doi = {10.11648/j.ajam.20200805.16},
      url = {https://doi.org/10.11648/j.ajam.20200805.16},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.ajam.20200805.16},
      abstract = {Pullback dynamics of nonautonomous dynamical systems has been considerably developed. However, it is still a tough job to study forward dynamics of nonautonomous dynamical systems, since forward attractors were only obtained in some particular cases. In the paper, under some reasonable conditions, it is shown that closing to a local pullback attractor, there is an approximate forward attractor. Specifically, let ϕ be a cocycle semiflow on a Banach space X with driving system θ on a base space P. Suppose that the base space P is compact and ϕ is uniformly asymptotically compact. Let A(∙) be a local pullback attractor with  being compact. We prove that every ε-extended neighborhood Aε(∙) of A(∙) will forward attract every bounded set B(∙) that is pullback attracted by A(∙). We then call Aε(∙) an approximate forward attractor of ϕ.},
     year = {2020}
    }
    

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    T1  - Local Approximate Forward Attractors of Nonautonomous Dynamical Systems
    AU  - Ailing Qi
    AU  - Xuewei Ju
    Y1  - 2020/09/25
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    JO  - American Journal of Applied Mathematics
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    AB  - Pullback dynamics of nonautonomous dynamical systems has been considerably developed. However, it is still a tough job to study forward dynamics of nonautonomous dynamical systems, since forward attractors were only obtained in some particular cases. In the paper, under some reasonable conditions, it is shown that closing to a local pullback attractor, there is an approximate forward attractor. Specifically, let ϕ be a cocycle semiflow on a Banach space X with driving system θ on a base space P. Suppose that the base space P is compact and ϕ is uniformly asymptotically compact. Let A(∙) be a local pullback attractor with  being compact. We prove that every ε-extended neighborhood Aε(∙) of A(∙) will forward attract every bounded set B(∙) that is pullback attracted by A(∙). We then call Aε(∙) an approximate forward attractor of ϕ.
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