Complutense University Library

Shape of global attractors in topological spaces

Giraldo, A. and Morón, Manuel A. and Romero Ruiz del Portal, Francisco and Rodríguez Sanjurjo, José Manuel (2005) Shape of global attractors in topological spaces. Nonlinear analysis-theory methods & applications, 60 (5). pp. 837-847. ISSN 0362-546X

[img] PDF
Restricted to Repository staff only until 31 December 2020.

277kB

Official URL: http://www.sciencedirect.com/science/article/pii/S0362546X04004274

View download statistics for this eprint

==>>> Export to other formats

Abstract

In this paper, we apply the notion and properties of compactly generated shape to study attractors in topological spaces.

Item Type:Article
Uncontrolled Keywords:Isolated invariant-sets; dynamical-systems; conley index; stability; flows; (semi)dynamical system; attractor; shape
Subjects:Sciences > Mathematics > Topology
ID Code:15273
References:

J.M. Ball, Stability theory for an extensible beam, J. Differential Equations 14 (1973) 399–418.

N.P. Bhatia, G.P. Szego, Stability Theory of Dynamical Systems, Springer, Berlin, 1970.

B.A. Bogatyi, V.I. Gutsu, On the structure of attracting compacta, Differentsial’nye Uravneniya 25 (1989) 907–909.

K. Borsuk, Theory of shape, Monografie Matematyczne, vol. 59, Polish Scientific Publishers, Warszawa, 1975.

T.A. Chapman, On some applications of infinite-dimensional manifolds to the theory of shape, Fund. Math. 76 (1972) 181–193.

T.A. Chapman, Shapes of finite-dimensional compacta, Fund. Math. 76 (1972) 261–276.

C.C. Conley, Isolated Invariant Sets and the Morse Index, CBMS, vol. 38, American Mathematical Society, Providence, RI, 1978.

J. Dydak, J. Segal, Shape Theory: An Introduction, Lecture Notes in Mathematics, vol. 688, Springer, Berlin, 1978.

R. Engelking, General Topology,Monografie Matematyczne, vol. 60, Polish Scientific Publishers,Warszawa, 1977.

B.M. Garay, Strong cellularity and global asymptotic stability, Fund. Math. 138 (1991) 147–154.

R. Geoghegan, R. Summerhill, Concerning the shapes of finite-dimensional compacta, Trans. AMS 179 (1973) 281–292.

A. Giraldo, R. Jimenez, M.A. Moron, F.R. Ruiz del Portal, J.M.R. Sanjurjo, Dissipativeness, global attractors and Borsuk’s shapes for metrizable spaces, preprint.

A. Giraldo, M.A. Moron, F.R. Ruiz del Portal, J.M.R. Sanjurjo, Some duality properties of non-saddle sets, Topology Appl. 113 (2001) 51–59.

A. Giraldo, J.M.R. Sanjurjo, On the global structure of invariant regions of flows with asymptotically stable attractors, Math. Z. 232 (1999) 739–746.

M. Gobbino, M. Sardella, On the connectedness of attractors for dynamical systems, J. Differential Equations 133 (1997) 1–14.

B. Gunther, Construction of differentiable flows with prescribed attractor, Topology Appl. 62 (1995) 87–91.

B. Gunther, J. Segal, Every attractor of a flow on a manifold has the shape of a finite polyhedron, Proc.AMS 119 (1993) 321–329.

J. Hale, Asymptotic Behaviour of Dissipative Systems, AMS Surveys and Monographics, vol. 25, American Mathematical Society, Providence, RI.

H.M. Hastings, Shape theory and dynamical systems, in: M.G. Markley, W. Perizzo (Eds.), The Structure of Attractors in Dynamical Systems, Lecture Notes in Mathematics, vol. 688, Springer, Berlin, 1978, pp. 150–160.

S.T. Hu, Theory of Retracts,Wayne State University Press, Detroit, 1965.

I. Ivanšic, R.B. Sher, A complement theorem for continua in a manifold, Topology Proc. 4 (1979) 437–452.

I. Ivanšic, R.B. Sher, G.A. Venema, Complement theorems beyond the trivial range, Illinois J. Math. 25 (1981) 209–220.

L. Kapitanski, I. Rodnianski, Shape and Morse theory of attractors, Comm. Pure Appl. Math. 53 (2000) 218–242.

S. Mardešic, Shapes for topological spaces, Gen. Topology Appl. 3 (1973) 265–282.

S. Mardešic, Absolute neighborhood retracts and shape theory, in: I.M. James (Ed.), History of Topology, North-Holland, Amsterdam, 1999, pp. 241–270.

S. Mardešic, J. Segal, Shape Theory, North-Holland, Amsterdam, 1982.

S. Mardešic, J. Segal, History of shape theory and its applications to general topology, in: C.E. Aull and R. Lowen (Eds.), Handbook of the History of GeneralTopology, vol. 3, Kluwer, Dordrecht, 2001, pp. 1145–1177.

A. Marzocchi, S.Z. Necca, Attractors for dynamical systems in topological spaces, Discrete Cont. Dynam. Sys. 8 (2002) 585–597.

M. Mrozek, Shape index and other indices of Conley type for local maps on locally compact Hausdorff spaces, Fund. Math. 145 (1) (1994) 15–37.

J.W. Robbin, D. Salamon, Dynamical systems, shape theory and the Conley index, Ergodic Theory Dynamical Systems 8 (1988) 375–393.

J.C. Robinson, Infinite dimensional dynamical systems, Texts in Applied Mathematics Series, Cambridge University Press, Cambridge, 2001.

J.C. Robinson, Global attractors: topology and finite dimensional dynamics, J. Dyn. Differential Equations 11 (1999) 557–581.

J.T. Rogers, The shape of a cross-section of the solution funnel of an ordinary differential equation, Illinois J. Math. 21 (1977) 420–426.

L. Rubin, J. Sanders, Compactly generated shape, Gen. Topology Appl. 4 (1974) 73–83.

D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. AMS. 291 (1985) 1–41.

J. Sanders, On the generalized and the H-shape theories, Duke Math. J. 49 (1973) 743–754.

J. Sanders, Shape groups on Hausdorff spaces, Glas. Math. 28 (1973) 297–304.

J.M.R. Sanjurjo, Multihomotopy, Cech spaces of maps and shape groups, Proc. London Math. Soc. 69 (1994) 330–344.

J.M.R. Sanjurjo, On the structure of uniform attractors, J. Math. Anal. Appl. 192 (1995) 519–528.

J.M.R. Sanjurjo, Morse equations and unstable manifolds of isolated invariant sets, Nonlinearity 16 (2003) 1435–1448.

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, American Mathematical Society, vol. 68, Springer, Berlin, 1988.

C. Tezer, Shift equivalence in homotopy, Math. Z. 210 (1992) 197–201.

W. Tucker, The Lorentz attractor exists, CR Acad. Sci. Paris 328 (I) (1999) 1197–1202

Deposited On:18 May 2012 08:57
Last Modified:06 Feb 2014 10:20

Repository Staff Only: item control page