Complutense University Library

Morse equations and unstable manifolds of isolated invariant sets

Rodríguez Sanjurjo, José Manuel (2003) Morse equations and unstable manifolds of isolated invariant sets. Nonlinearity, 16 (4). pp. 1435-1448. ISSN 0951-7715

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


Official URL:

View download statistics for this eprint

==>>> Export to other formats


We describe a new way of obtaining the Morse equations of a Morse decomposition of an isolated invariant set. This is achieved through a filtration of truncated unstable manifolds associated with the decomposition. The results in the paper make it possible to calculate the Morse equations (and also the Conley index) in many interesting situations without using index pairs. We also study the intrinsic topology of the unstable manifold and obtain new duality properties of the cohomological Conley index.

Item Type:Article
Uncontrolled Keywords:Index theory, Morse-Conley indices; Strange attractors, chaotic dynamics; Stability theory
Subjects:Sciences > Mathematics > Geometry
Sciences > Mathematics > Topology
ID Code:16824

Bhatia N P and Szeg¨o G P 1970 Stability Theory of Dynamical Systems (Grundlehren der Mat. Wiss. 161) (Berlin: Springer)

Borsuk K 1975 Theory of Shape, Monografie Matematyczne, Tom 59 (Mathematical Monographs) vol 59 (Warsaw: PWN-Polish Scientific Publishers)

Conley C 1978 Isolated invariant sets and the Morse index CBMS Regional Conference Series in Mathematics vol 38 (Providence, RI: American Mathematical Society)

Conley C and Zehnder E 1984 Morse-type index theory for flows and periodic solutions for Hamiltonian equations Commun. Pure Appl. Math. 37 207–53

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

Giraldo A and Sanjurjo J M R 1999 On the global structure of invariant regions of flows with asymptotically stable attractors Mathematische Zeitschrift 232 739-746

Guckenheimer J and Holmes P 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Berlin: Springer)

Gunther B and Segal J 1993 Every attractor of a flow on a manifold has the shape of a finite polyhedron Proc. AMS 119 321–9

Kapitanski L and Rodnianski I 2000 Shape and Morse theory of attractors Comm. Pure Appl. Math. 53 218–42

Lorenz E N 1963 Deterministic non-period flows J. Atmos. Sci. 20 130–41

Mardesic S 1974 Pairs of compacta and trivial shape Trans. AMS 189 329–36

Mardesic S and Segal J 1982 Shape Theory. The Inverse System Approach (North-Holland Mathematical Library) vol 26 (Amsterdam: North-Holland)

Mawhin J and Willem M 1989 Critical Point Theory and Hamiltonian Systems (New York: Springer)

McCord C 1988 Mappings and homological properties in the Conley index theory Ergodic Theory Dynam. Syst. 8 (Charles Conley Memorial Volume) 175–98

McCord C 1992 Poincare–Lefschetz duality for the homology Conley index Trans. AMS 329 233–52

Robbin J W and Salamon D 1988 Dynamical systems, shape theory and the Conley index Ergodic Theory Dynam. Syst. 8 (Charles Conley Memorial Volume) 375–93

Rybakowski K P 1987 The Homotopy Index and Partial Differential Equations, Universitext (Berlin: Springer)

Rybakowski K P and Zehnder E 1985 AMorse equation in Conley’s index theory for semiflows on metric spaces Ergodic Theory Dynam. Syst. 5 123–43

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

Sanjurjo J M R 1994 Multihomotopy, Cech spaces of loops and shape groups Proc. London Math. Soc. 69 330–44

Sanjurjo J M R 1995 On the structure of uniform attractors J. Math. Anal. Appl. 192 519–28

Spanier E H 1966 Algebraic Topology (New York: McGraw-Hill)

Sparrow C 1982 The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors (Berlin: Springer)

Tucker W 1999 The Lorenz attractor exists C. R. Acad. Sci. Paris t. 328, Serie I 1197–202

Deposited On:23 Oct 2012 09:04
Last Modified:07 Feb 2014 09:36

Repository Staff Only: item control page