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Stable and unstable minimal attractors Konstantin Athanassopoulos
 

Summary: Stable and unstable minimal attractors
Konstantin Athanassopoulos
Department of Mathematics, University of Crete, GR-71409 Iraklion, Greece
1. Introduction
Two fundamental problems in the qualitative theory of ordinary differential equations
dynamical systems are (a) the study of the topology and dynamics in compact invariant
sets of a continuous flow, and in particular compact minimal sets, and (b) the description
of the dynamics around such a set. We will be concerned with the connection between
these two problems, that is how the complexity of a compact minimal set affects the
behaviour of the flow around it.
The simplest behavior occurs near an asymptotically stable compact invariant set.
Let (t)tR be of a continuous flow on a separable, locally compact, metrizable space M.
The positive limit set of x M is the closed, invariant set
L+
(x) = {y M : tn (x) y for some tn +}.
Obviously, L+(t(x)) = L+(x) for every t R. Let A M be a compact invariant set.
The invariant set
W+
(A) = {x M : = L+
(x) A}

  

Source: Athanassopoulos, Konstantin - Department of Mathematics, University of Crete

 

Collections: Mathematics