Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T11:34:05.907Z Has data issue: false hasContentIssue false
  • English
  • Français

Tubular neighborhoods and continuation of Morse decompositions

Published online by Cambridge University Press:  03 July 2014

M. C. CARBINATTO  and
K. P. RYBAKOWSKI
M. C. CARBINATTO
Affiliation:
Departamento de Matemática, ICMC-USP, Caixa Postal 668, 13.560-970 São Carlos SP, Brazil email [email protected]
K. P. RYBAKOWSKI
Affiliation:
Universität Rostock, Institut für Mathematik, Ulmenstrasse 69, Haus 3, 18057 Rostock, Germany email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove a continuation result for Morse decompositions under tubular singular semiflow perturbations, which generalizes a corresponding result from Carbinatto and Rybakowski [Morse decompositions in the absence of uniqueness, II. Topol. Methods Nonlinear Anal.22 (2003), 15–51] and is applicable to cases in which the phase space of the perturbed semiflow is not necessarily homeomorphic to a product of metric spaces having as a factor the phase space of the limiting semiflow. We apply this result to singularly perturbed second-order differential equations on differential manifolds.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 7 , October 2015 , pp. 2053 - 2079
Copyright
© Cambridge University Press, 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Carbinatto, M. C. and Rybakowski, K. P.. Morse decompositions in the absence of uniqueness. Topol. Methods Nonlinear Anal. 18 (2001), 205–242.CrossRefGoogle Scholar
Carbinatto, M. C. and Rybakowski, K. P.. On a general Conley index continuation principle for singular perturbation problems. Ergod. Th. & Dynam. Sys. 22 (2002), 729–755.Google Scholar
Carbinatto, M. C. and Rybakowski, K. P.. Morse decompositions in the absence of uniqueness, II. Topol. Methods Nonlinear Anal. 22 (2003), 15–51.CrossRefGoogle Scholar
Carbinatto, M. C. and Rybakowski, K. P.. Nested index filtrations and continuation of the connection matrix. J. Differential Equations 207 (2004), 458–488.CrossRefGoogle Scholar
Carbinatto, M. C. and Rybakowski, K. P.. Continuation of the connection matrix in singular perturbation problems. Ergod. Th. & Dynam. Sys. 26 (2006), 1021–1059.CrossRefGoogle Scholar
Carbinatto, M. C. and Rybakowski, K. P.. On convergence and compactness in parabolic problems with globally large diffusion and nonlinear boundary conditions. Topol. Methods Nonlinear Anal. 40 (2012), 1–28.Google Scholar
Carbinatto, M. C. and Rybakowski, K. P.. Continuation of the connection matrix for singularly perturbed hyperbolic equations. Fund. Math. 196 (2007), 253–273.CrossRefGoogle Scholar
Carbinatto, M. C. and Rybakowski, K. P.. Conley index and tubular neighborhoods. J. Differential Equations 254 (2013), 933–959.CrossRefGoogle Scholar
Conley, C. C.. Isolated Invariant Sets and the Morse Index (Conference Board of the Mathematical Sciences, 38). American Mathematical Society, Providence, RI, 1978.CrossRefGoogle Scholar
Ćwiszewski, A. and Rybakowski, K. P.. Singular dynamics of the strongly damped beam equation. J. Differential Equations 247 (2009), 3202–3233.CrossRefGoogle Scholar
Furi, M.. Second order differential equations on manifolds and forced oscillations (notes by M. Spadini). Proceedings of the Conference on Topological Methods in Differential Equations and Inclusions (Université de Montréal, 1994) (NATO ASI Series C). Kluwer, Dordrecht, 1995.Google Scholar
Franzosa, R.. The connection matrix theory for Morse decompositions. Trans. Amer. Math. Soc. 311 (1989), 561–592.CrossRefGoogle Scholar
Franzosa, R.. The continuation theory for Morse decompositions and connection matrices. Trans. Amer. Math. Soc. 310 (1988), 781–803.CrossRefGoogle Scholar
Franzosa, R. D. and Mischaikow, K.. The connection matrix theory for semiflows on (not necessarily locally compact) metric spaces. J. Differential Equations 71 (1988), 270–287.CrossRefGoogle Scholar
Izydorek, M. and Rybakowski, K. P.. Multiple solutions of strongly indefinite elliptic systems via a Galerkin-type Conley index theory. Fund. Math. 176 (2003), 233–249.CrossRefGoogle Scholar
Kaczyński, T.. Conley index for set-valued maps: from theory to computation. Banach Center Publ. 47 (1999), 57–65.CrossRefGoogle Scholar
Kaczyński, T. and Mrozek, M.. Conley index for discrete multivalued dynamical systems. Topology Appl. 65 (1995), 83–96.CrossRefGoogle Scholar
McGehee, R.. Attractors for closed relations on compact Hausdorff spaces. Indiana Univ. Math. J. 41 (1992), 1165–1209.CrossRefGoogle Scholar
Mrozek, M.. A cohomological index of Conley type for multi-valued admissible flows. J. Differential Equations 84 (1990), 15–51.CrossRefGoogle Scholar
Roxin, E.. Stability in general control systems. J. Differential Equations 1 (1965), 115–150.CrossRefGoogle Scholar
Rybakowski, K. P.. On the homotopy index for infinite-dimensional semiflows. Trans. Amer. Math. Soc. 269 (1982), 351–382.CrossRefGoogle Scholar
Rybakowski, K. P.. The Morse-index, repeller–attractor pairs and the connection index for semiflows on noncompact spaces. J. Differential Equations 47 (1983), 66–98.CrossRefGoogle Scholar
Rybakowski, K. P.. The Homotopy Index and Partial Differential Equations. Springer, Berlin, 1987.CrossRefGoogle Scholar
Rybakowski, K. P.. Conley index and singularly perturbed hyperbolic equations. Topol. Methods Nonlinear Anal. 22 (2003), 203–244.CrossRefGoogle Scholar
Rybakowski, K. P. and Zehnder, E.. On a Morse equation in Conley’s index theory for semiflows on metric spaces. Ergod. Th. & Dynam. Sys. 5 (1985), 123–143.CrossRefGoogle Scholar