Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Construction of nonautonomous forward attractors


Authors: Peter E. Kloeden and Thomas Lorenz
Journal: Proc. Amer. Math. Soc. 144 (2016), 259-268
MSC (2010): Primary 34B45, 37B55; Secondary 37C70
DOI: https://doi.org/10.1090/proc/12735
Published electronically: May 28, 2015
MathSciNet review: 3415594
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Autonomous systems depend only on the elapsed time, so their attractors and limit sets exist in current time. Similarly, the pullback limit defines a component set of a nonautonomous pullback attractor at each instant of current time. The forward limit defining a nonautonomous forward attractor is different as it is the limit to the asymptotically distant future. In particular, the limiting objects forward in time do not have the same dynamical meaning in current time as in the autonomous or pullback cases. Nevertheless, the pullback limit taken within a positively invariant family of compact subsets allows the component set of a forward attractor to be constructed at each instant of current time. Every forward attractor has such a positively invariant family of compact subsets, which ensures that the component sets of a forward attractor can be constructed in this way. It is, however, only a necessary condition and not sufficient for the constructed family of subsets to be a forward attractor. The analysis here is presented in the state space $ \mathbb{R}^d$ to focus on the dynamical essentials rather than on functional analytical technicalities; in particular, those concerning asymptotic compactness properties.


References [Enhancements On Off] (What's this?)

  • [1] A. N. Carvalho, J. A. Langa, and J. C. Robinson, Attractors of infinite dimensional nonautonomous dynamical systems, Springer-Verlag, Berlin, 2012.
  • [2] Alexandre N. Carvalho, José A. Langa, James C. Robinson, and Antonio Suárez, Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system, J. Differential Equations 236 (2007), no. 2, 570-603. MR 2322025 (2008e:37075), https://doi.org/10.1016/j.jde.2007.01.017
  • [3] D. N. Cheban, P. E. Kloeden, and B. Schmalfuß, The relationship between pullback, forward and global attractors of nonautonomous dynamical systems, Nonlinear Dyn. Syst. Theory 2 (2002), no. 2, 125-144. MR 1989935 (2004e:34090)
  • [4] Vladimir V. Chepyzhov and Mark I. Vishik, Attractors for equations of mathematical physics, American Mathematical Society Colloquium Publications, vol. 49, American Mathematical Society, Providence, RI, 2002. MR 1868930 (2003f:37001c)
  • [5] Igor Chueshov, Monotone random systems theory and applications, Lecture Notes in Mathematics, vol. 1779, Springer-Verlag, Berlin, 2002. MR 1902500 (2003d:37072)
  • [6] Constantine M. Dafermos, An invariance principle for compact processes, J. Differential Equations 9 (1971), 239-252; erratum, ibid. 10 (1971), 179-180. MR 0291596 (45 #687)
  • [7] Jack K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR 941371 (89g:58059)
  • [8] J. Kato, A. A. Martynyuk, and A. A. Sheshakov, Stability of motion of nonautonomous systems, Gordon & Breach Publ., London, (1996).
  • [9] Peter E. Kloeden and Pedro Marín-Rubio, Negatively invariant sets and entire solutions, J. Dynam. Differential Equations 23 (2011), no. 3, 437-450. MR 2836645, https://doi.org/10.1007/s10884-010-9196-8
  • [10] Peter E. Kloeden, Christian Pötzsche, and Martin Rasmussen, Limitations of pullback attractors for processes, J. Difference Equ. Appl. 18 (2012), no. 4, 693-701. MR 2905291, https://doi.org/10.1080/10236198.2011.578070
  • [11] P. E. Kloeden, C. Pötzsche, and M. Rasmussen, Discrete-time nonautonomous dynamical systems, Stability and bifurcation theory for non-autonomous differential equations, Lecture Notes in Math., vol. 2065, Springer, Heidelberg, 2013, pp. 35-102. MR 3203916, https://doi.org/10.1007/978-3-642-32906-7_2
  • [12] Peter E. Kloeden and Martin Rasmussen, Nonautonomous dynamical systems, Mathematical Surveys and Monographs, vol. 176, American Mathematical Society, Providence, RI, 2011. MR 2808288 (2012g:37038)
  • [13] Peter E. Kloeden and Jacson Simsen, Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents, J. Math. Anal. Appl. 425 (2015), no. 2, 911-918. MR 3303900, https://doi.org/10.1016/j.jmaa.2014.12.069
  • [14] J. P. LaSalle, The stability of dynamical systems, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976. With an appendix: ``Limiting equations and stability of nonautonomous ordinary differential equations'' by Z. Artstein; Regional Conference Series in Applied Mathematics. MR 0481301 (58 #1426)
  • [15] Christian Pötzsche, Geometric theory of discrete nonautonomous dynamical systems, Lecture Notes in Mathematics, vol. 2002, Springer-Verlag, Berlin, 2010. MR 2680867 (2012a:37003)
  • [16] Martin Rasmussen, Attractivity and bifurcation for nonautonomous dynamical systems, Lecture Notes in Mathematics, vol. 1907, Springer, Berlin, 2007. MR 2327977 (2008k:37040)
  • [17] Emilio Roxin, Stability in general control systems, J. Differential Equations 1 (1965), 115-150. MR 0201755 (34 #1637)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 34B45, 37B55, 37C70

Retrieve articles in all journals with MSC (2010): 34B45, 37B55, 37C70


Additional Information

Peter E. Kloeden
Affiliation: School of Mathematics and Statistics, Huazhong University of Science & Technology, Wuhan 430074, People’s Republic of China
Email: kloeden@math.uni-frankfurt.de

Thomas Lorenz
Affiliation: Applied Mathematics, RheinMain University of Applied Sciences, 65197 Wiesbaden, Germany
Email: thomas.lorenz@hs-rm.de

DOI: https://doi.org/10.1090/proc/12735
Keywords: Nonautonomous dynamical system, 2-parameter semi-group, pullback attractor, forward attractor, omega limit points
Received by editor(s): October 18, 2014
Received by editor(s) in revised form: December 14, 2014
Published electronically: May 28, 2015
Additional Notes: The first author was partially supported by the DFG grants KL 1203/7-1 and LO 273/5-1, the Spanish Ministerio de Economía y Competitividad project MTM2011-22411, the Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía) under the Ayuda 2009/FQM314, and the Proyecto de Excelencia : P12-FQM-1492
The second author was partially supported by the DFG grants KL 1203/7-1 and LO 273/5-1.
Communicated by: Yingfei Yi
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society