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 Tr. Mat. Inst. Steklova, 2012, Volume 277, Pages 91–100 (Mi tm3385)

Relatively unstable attractors

Yu. S. Ilyashenkoabcde, I. S. Shilinf

a Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia
b Independent University of Moscow, Moscow, Russia
c National Research University Higher School of Economics, Moscow, Russia
d Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia
e Cornell University, Ithaca, NY, USA
f Moscow State University, Moscow, Russia

Abstract: There are different non-equivalent definitions of attractors in the theory of dynamical systems. The most common are two definitions: the maximal attractor and the Milnor attractor. The maximal attractor is by definition Lyapunov stable, but it is often in some ways excessive. The definition of Milnor attractor is more realistic from the physical point of view. The Milnor attractor can be Lyapunov unstable though. One of the central problems in the theory of dynamical systems is the question of how typical such a phenomenon is. This article is motivated by this question and contains new examples of so-called relatively unstable Milnor attractors. Recently I. Shilin has proved that these attractors are Lyapunov stable in the case of one-dimensional fiber under some additional assumptions. However, the question of their stability in the case of multidimensional fiber is still an open problem.

 Funding Agency Grant Number National Science Foundation 0700973 Russian Foundation for Basic Research 10-01-00739-a10-01-93115-NTsNIL_a This work was supported in part by the NSF (project no. 0700973), Russian Foundation for Basic Research (project no. 10-01-00739-a), and RFBR-CNRS (project no. 10-01-93115-NTsNIL_a).

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English version:
Proceedings of the Steklov Institute of Mathematics, 2012, 277, 84–93

Bibliographic databases:

UDC: 517.938

Citation: Yu. S. Ilyashenko, I. S. Shilin, “Relatively unstable attractors”, Mathematical control theory and differential equations, Collected papers. In commemoration of the 90th anniversary of Academician Evgenii Frolovich Mishchenko, Tr. Mat. Inst. Steklova, 277, MAIK Nauka/Interperiodica, Moscow, 2012, 91–100; Proc. Steklov Inst. Math., 277 (2012), 84–93

Citation in format AMSBIB
\Bibitem{IlyShi12} \by Yu.~S.~Ilyashenko, I.~S.~Shilin \paper Relatively unstable attractors \inbook Mathematical control theory and differential equations \bookinfo Collected papers. In commemoration of the 90th anniversary of Academician Evgenii Frolovich Mishchenko \serial Tr. Mat. Inst. Steklova \yr 2012 \vol 277 \pages 91--100 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm3385} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3052266} \elib{http://elibrary.ru/item.asp?id=17759400} \transl \jour Proc. Steklov Inst. Math. \yr 2012 \vol 277 \pages 84--93 \crossref{https://doi.org/10.1134/S0081543812040074} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000309232900007} \elib{http://elibrary.ru/item.asp?id=23960339} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84904045443} 

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This publication is cited in the following articles:
1. Yu. S. Ilyashenko, “Multidimensional Bony Attractors”, Funct. Anal. Appl., 46:4 (2012), 239–248
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