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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
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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.


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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

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