### Impacto

Sánchez Gabites, Jaime Jorge and Rodríguez Sanjurjo, José Manuel
(2007)
*On the topology of the boundary of a basin of attraction.*
Proceedings of the American Mathematical Society, 135
(12).
pp. 4087-4098.
ISSN 0002-9939

PDF
Restringido a Repository staff only hasta 31 December 2020. 491kB |

Official URL: http://www.ams.org/journals/proc/2007-135-12/S0002-9939-07-08972-1/S0002-9939-07-08972-1.pdf

## Abstract

Suppose phi : M x R -> M is a continuous flow on a locally compact metrizable space M and K is an ( asymptotically stable) attractor. Let D =partial derivative A( K) be the boundary of the basin of attraction of K. In the present paper it will be shown how the Conley index of D plays an important role in determining the topological nature of D and allows one to obtain information about the global dynamics of phi in M.

Item Type: | Article |
---|---|

Subjects: | Sciences > Mathematics > Geometry Sciences > Mathematics > Topology |

ID Code: | 16805 |

References: | J. Auslander, N. P. Bhatia, and P. Seibert, Attractors in dynamical systems, Bol. Soc. Mat. Mexicana (2) 9 (1964), 55–66. N. P. Bhatia and G. P. Szego, Stability theory of dynamical systems, Die Grundlehren der mathematischen Wissenschaften, Band 161, Springer-Verlag, 1970. S. A. Bogatyı and V. I. Gutsu, On the structure of attracting compacta, Differentsialnye Uravneniya 25 (1989), no. 5, 907–909, 920. K. Borsuk, Theory of shape, Monografie Matematyczne, Tom 59, Panstwowe Wydawnictwo Naukowe, 1975 R. C. Churchill, Isolated invariant sets in compact metric spaces, J. Diff. Eq. 12 (1972), 330–352. C. Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, 38, American Mathematical Society, 1978 C. Conley and R. Easton, Isolated invariant sets and isolating blocks, Trans. Amer. Math. Soc. 158 (1971), no. 35–61. J. Dydak and J. Segal, Shape theory. An introduction, Lecture Notes in Mathematics, 688, Springer, 1978. R. W. Easton, Geometric methods for discrete dynamical systems, Oxford Engineering Science Series, 50, Oxford University Press, 1998. A. Giraldo, M. A. Mor´on, F. R. Ruiz del Portal, and J. M. R. Sanjurjo, Shape of global attractors in topological spaces, Nonlinear Anal. 60 (2005), no. 5, 837–847 A. Giraldo and J. M. R. Sanjurjo, On the global structure of invariant regions of flows with asymptotically stable attractors, Math. Z. 232 (1999), no. 4, 739–746. B. Gunther and J. Segal, Every attractor of a flow on a manifold has the shape of a finite polyhedron, Proc. Amer. Math. Soc. 119 (1993), no. 1, 321–329. S. Hu, Theory of retracts, Wayne State University Press, 1965. A. Kadlof, On the shape of pointed compact connected subsets of E3, Fund. Math. 115 (1983), no. 3. S. Mardesic and J. Segal, Shape theory. The inverse system approach, North–Holland Mathematical Library, 26, North–Holland Publishing Co., 1982. J. W. Milnor, Topology from the differentiable viewpoint, The University Press of Virginia, 1965. H. E. Nusse and J. A. Yorke, Characterizing the basins with the most entangled boundaries, Ergodic Theory Dynam. Systems 23 (2003), no. 3, 895–906. D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985), no. 1, 1–41 J. M. R. Sanjurjo, Multihomotopy, Cech spaces of loops and shape groups, Proc. London Math. Soc. (3) 69 (1994), no. 2, 330–344. _ On the structure of uniform attractors, J. Math. Anal. Appl. 192 (1995), no. 2, 519–528. E. H. Spanier, Algebraic topology, McGraw Hill Book Co., 1966 J. E. West, Mapping Hilbert cube manifolds to ANR’s: a solution of a conjecture of Borsuk, Ann. Math. (2) 106 (1977), 1–18. |

Deposited On: | 22 Oct 2012 10:48 |

Last Modified: | 07 Feb 2014 09:36 |

Repository Staff Only: item control page