International Conference «Mathematical and Informational Technologies, MIT-2011»
(IX Conference «Computational and Informational Technologies for Science,
Engineering and Education»)

Vrnjacka Banja, Serbia, August, 27–31, 2011

Budva, Montenegro, August, 31 – September, 5, 2011

Shekutkovski N.  

Intrinsic shape, attractors and Morse decomposition

     One of the most important problems in dynamical systems concerns the asymptotic behavior of trajectories as time goes to plus or minus infinity. Limit sets are fundamental tools for this problem. Several results will be presented concerning shape of limit sets. Intrinsic shape recently introduced by the author in [5], throws a new light to this area of study of dynamical systems. The intrinsic shape is a very appropriate tool, since it uses only the properties of the existing phase space X of the dynamical system f: XxR->X, not introducing external spaces.
     For a family of disjoint compact invariant subsets {K1,K2,…,Kn} of the phase space X, a Lyapunov function is a continuous function L: XxR->X such that: L(f(x,t) < L(x) for all t>0 and all x not belonging to any of compact sets K1,K2,…,Kn , and L(K1)=C1, L(K2)=C2, … , L(Kn)=Cn (C1, C2,…, Cn constants).
      Then, the family {K1,K2,…,Kn} is a Morse decomposition of X.
      Every two element Morse decomposition {K1, K2} is in fact an attractor-repelor pair. In this case there are several results about the shape of attractor [1], [2], [3] and [4]. Based on the new intrinsic approach to shape in the paper [5], for arbitrary Morse decomposition {K1, K2,…, Kn} of a given flow f, it is proven that there exist compact neighborhoods of U1 of K1 having the same shape as K1, and neighbourhoods U2 of K2, … , Un of Kn, having the same shape, respectively.

[1] С.А Богатый, В. И. Гуцу, О структуре притягивающих компактов, Дифференциальные уравнения 25 (1985), 907-909.
[2] B. Gunther, J. Segal , “Every attractor of a flow on a manifold has the shape of a finite polyhedron”, Proceedings of the American Mathematical Society, Volume 119.
[3] Rodnianski, Kapitanski, Shape and Morse theory of attractors, Communications On Pure and Applied Mathematics , 53 (2000), no. 2, 218-242.
[4] J. M. R. Sanjurjo, On the structure of uniform attractors, J. Math. Anal. Appl. 192 (1995), 519-528.
[5] N. Shekutkovski, Intrinsic definition of strong shape for compact metric spaces, Topology Proceedings 39 (2012), 27 - 39.

Abstracts file: abstract_InSh&Attr.doc

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