## Construction of nonautonomous forward attractors

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- by Peter E. Kloeden and Thomas Lorenz
- Proc. Amer. Math. Soc.
**144**(2016), 259-268 - DOI: https://doi.org/10.1090/proc/12735
- Published electronically: May 28, 2015
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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.## References

- A. N. Carvalho, J. A. Langa, and J. C. Robinson,
*Attractors of infinite dimensional nonautonomous dynamical systems,*Springer-Verlag, Berlin, 2012. - 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**, DOI 10.1016/j.jde.2007.01.017 - 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** - 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**, DOI 10.1051/cocv:2002056 - Igor Chueshov,
*Monotone random systems theory and applications*, Lecture Notes in Mathematics, vol. 1779, Springer-Verlag, Berlin, 2002. MR**1902500**, DOI 10.1007/b83277 - Constantine M. Dafermos,
*An invariance principle for compact processes*, J. Differential Equations**9**(1971), 239–252; erratum, ibid. 10 (1971), 179–180. MR**291596**, DOI 10.1016/0022-0396(71)90078-7 - Jack K. Hale,
*Asymptotic behavior of dissipative systems*, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR**941371**, DOI 10.1090/surv/025 - J. Kato, A. A. Martynyuk, and A. A. Sheshakov,
*Stability of motion of nonautonomous systems*, Gordon & Breach Publ., London, (1996). - 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**, DOI 10.1007/s10884-010-9196-8 - 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**, DOI 10.1080/10236198.2011.578070 - 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**, DOI 10.1007/978-3-642-32906-7_{2} - Peter E. Kloeden and Martin Rasmussen,
*Nonautonomous dynamical systems*, Mathematical Surveys and Monographs, vol. 176, American Mathematical Society, Providence, RI, 2011. MR**2808288**, DOI 10.1090/surv/176 - 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**, DOI 10.1016/j.jmaa.2014.12.069 - J. P. LaSalle,
*The stability of dynamical systems*, Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976. With an appendix: “Limiting equations and stability of nonautonomous ordinary differential equations” by Z. Artstein. MR**0481301** - Christian Pötzsche,
*Geometric theory of discrete nonautonomous dynamical systems*, Lecture Notes in Mathematics, vol. 2002, Springer-Verlag, Berlin, 2010. MR**2680867**, DOI 10.1007/978-3-642-14258-1 - Martin Rasmussen,
*Attractivity and bifurcation for nonautonomous dynamical systems*, Lecture Notes in Mathematics, vol. 1907, Springer, Berlin, 2007. MR**2327977** - Emilio Roxin,
*Stability in general control systems*, J. Differential Equations**1**(1965), 115–150. MR**201755**, DOI 10.1016/0022-0396(65)90015-X

## Bibliographic Information

**Peter E. Kloeden**- Affiliation: School of Mathematics and Statistics, Huazhong University of Science & Technology, Wuhan 430074, People’s Republic of China
- MR Author ID: 102990
- 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
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**144**(2016), 259-268 - MSC (2010): Primary 34B45, 37B55; Secondary 37C70
- DOI: https://doi.org/10.1090/proc/12735
- MathSciNet review: 3415594