|
Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2012, Volume 277, Pages 91–100
(Mi tm3385)
|
|
|
|
This article is cited in 2 scientific papers (total in 2 papers)
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.
Received in December 2011
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, Trudy Mat. Inst. Steklova, 277, MAIK Nauka/Interperiodica, Moscow, 2012, 91–100; Proc. Steklov Inst. Math., 277 (2012), 84–93
Linking options:
https://www.mathnet.ru/eng/tm3385 https://www.mathnet.ru/eng/tm/v277/p91
|
Statistics & downloads: |
Abstract page: | 431 | Full-text PDF : | 94 | References: | 81 |
|