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1996-1997 | 65 | 3 | 203-211
Tytuł artykułu

An attraction result and an index theorem for continuous flows on $ℝ^n × [0,∞)$

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We study the behavior of a continuous flow near a boundary. We prove that if φ is a flow on $E = ℝ^{n+1}$ for which $∂E = ℝ^n × {0}$ is an invariant set and S ⊂ ∂E is an isolated invariant set, with non-zero homological Conley index, then there exists an x in E\∂E such that either α(x) or ω(x) is in S. We also prove an index theorem for a flow on $ℝ^n × [0,∞)$.
Słowa kluczowe
Kategorie tematyczne
Rocznik
Tom
65
Numer
3
Strony
203-211
Opis fizyczny
Daty
wydano
1997
otrzymano
1995-01-30
poprawiono
1995-05-06
Twórcy
  • Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland
Bibliografia
  • [Ca-Ga] A. Capietto and B. M. Garay, Saturated invariant sets and boundary behavior of differential systems, J. Math. Anal. Appl. 176 (1993), 166-181.
  • [Ch] R. C. Churchill, Isolated invariant sets in compact metric spaces, J. Differential Equations 12 (1972), 330-352.
  • [Co] C. C. Conley, Isolated invariant sets and the Morse index, CBMS Regional Conf. Ser. in Math. 38, Amer. Math. Soc., Providence, R.I., 1978.
  • [Do] A. Dold, Lectures on Algebraic Topology, Springer, Berlin, 1972.
  • [Ho] J. Hofbauer, Saturated equilibria, permanence, and stability for ecological systems, in: Mathematical Ecology, Proc. Trieste 1986, L. J. Gross, T. G. Hallam and S. A. Levin (eds.), World Scientific, 625-642.
  • [Ho1] J. Hofbauer, An index theorem for dissipative semiflows, Rocky Mountain J. Math. 20 (1990), 1017-1031.
  • [Ho-Si] J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems, Cambridge Univ. Press, Cambridge, 1988.
  • [Mr1] M. Mrozek, Leray functor and cohomological Conley index for discrete dynamical systems, Trans. Amer. Math. Soc. 318 (1990), 149-178.
  • [Mr2] M. Mrozek, The Conley index on compact ANR's is of finite type, Results Math. 18 (1990), 306-313.
  • [Mr3] M. Mrozek, Open index pairs, the fixed point index and rationality of zeta function, Ergodic Theory Dynam. Systems 10 (1990), 555-564.
  • [Mr4] M. Mrozek, Index pairs and the fixed point index for semidynamical systems with discrete time, Fund. Math. 133 (1989), 177-194.
  • [Mr-Srz] M. Mrozek and R. Srzednicki, On time-duality of the Conley index, Results Math. 24 (1993), 161-167.
  • [Ryb] K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer, Berlin, 1987.
  • [Srz] R. Srzednicki, On rest points of dynamical systems, Fund. Math. 126 (1985), 69-81.
  • [Sz] A. Szymczak, The Conley index and symbolic dynamics, Topology 35 (1996), 287-299.
Typ dokumentu
Bibliografia
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