### Non-attracting attractors

Yulij S. Ilyashenko,

Independent University of Moscow,
Steklov Institute of Mathematics, Lomonosov University of Moscow
and Cornell

One of the major problems in the theory of dynamical systems is the

study of the limit behavior of solutions. It is a general belief

that after a long time delay the observer will see the orbits that

belong to this or that type of attractor. Therefore, knowledge of

the attractor of the system predicts the long time behavior of

solutions.

In the present talk we develop an opposite point of view. Namely, we

describe dynamical systems whose attractors have a large part which

is in a sense unobservable. This motivated a notion of $\varepsilon

$-attractor. It is a set (not necessary uniquely defined ) near

which almost all the orbits spend in average more than

$1-\varepsilon $ part of the future time. We discuss the effect of

drastic non-coincidence of actual attractor and $\varepsilon

$-attractor. For $\varepsilon$ sufficiently small, like $10^{-30}$,

the difference between actual attractors and $\varepsilon$

attractors is unobservable in the computer and physical experiments.

Therefore, $\varepsilon$-attractors with small $\varepsilon$ have a

chance to replace actual attractors in applications. Theorems and

conjectures on the subject will be presented.

This is a joint work with Andrei Negut, senior student of Princeton

University.

#### Colloquium, Department of Mathematics, Rice University

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