Local Approximate Forward Attractors of Nonautonomous Dynamical Systems

Ailing Qi; Xuewei Ju

doi:doi:10.11648/j.ajam.20200805.16

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Local Approximate Forward Attractors of Nonautonomous Dynamical Systems

Ailing Qi, Xuewei Ju

Published in American Journal of Applied Mathematics (Volume 8, Issue 5)

Received: 6 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 A^ε(∙) of A(∙) will forward attract every bounded set B(∙) that is pullback attracted by A(∙). We then call A^ε(∙) an approximate forward attractor of ϕ.

Published in	American Journal of Applied Mathematics (Volume 8, Issue 5)
DOI	10.11648/j.ajam.20200805.16
Page(s)	278-283
Creative Commons	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), 2020. 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.
[4]	V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonmous dynamical systems and their dimension, J. Math. Pures Appl., 73, (1994), 279--333.
[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.
[17]	M. I. Vishik, Asymptotic Behavior of Solutions of Evlutionary Equations, Cambridge University Press, Cambriage, England, 1992.
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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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ACS 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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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://article.sciencepublishinggroup.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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TY  - JOUR
T1  - Local Approximate Forward Attractors of Nonautonomous Dynamical Systems
AU  - Ailing Qi
AU  - Xuewei Ju
Y1  - 2020/09/25
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200805.16
DO  - 10.11648/j.ajam.20200805.16
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 278
EP  - 283
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200805.16
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.
VL  - 8
IS  - 5
ER  -

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Author Information

Ailing Qi

Department of Mathematics, Civil Aviation University of China, Tianjin, China
Xuewei Ju

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

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APA Style

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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ACS 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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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

Copy | Download

@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://article.sciencepublishinggroup.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}
}

Copy | Download

TY  - JOUR
T1  - Local Approximate Forward Attractors of Nonautonomous Dynamical Systems
AU  - Ailing Qi
AU  - Xuewei Ju
Y1  - 2020/09/25
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200805.16
DO  - 10.11648/j.ajam.20200805.16
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 278
EP  - 283
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200805.16
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.
VL  - 8
IS  - 5
ER  -

Copy | Download