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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Morse decompositions of nonautonomous dynamical systems


Author: Martin Rasmussen
Journal: Trans. Amer. Math. Soc. 359 (2007), 5091-5115
MSC (2000): Primary 34D05, 37B25, 37B55, 37C70; Secondary 34D08
Published electronically: April 24, 2007
MathSciNet review: 2320661
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Abstract: The global asymptotic behavior of dynamical systems on compact metric spaces can be described via Morse decompositions. Their components, the so-called Morse sets, are obtained as intersections of attractors and repellers of the system. In this paper, new notions of attractor and repeller for nonautonomous dynamical systems are introduced which are designed to establish nonautonomous generalizations of the Morse decomposition. The dynamical properties of these decompositions are discussed, and nonautonomous Lyapunov functions which are constant on the Morse sets are constructed explicitly. Moreover, Morse decompositions of one-dimensional and linear systems are studied.


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Additional Information

Martin Rasmussen
Affiliation: Department of Mathematics, University of Augsburg, D-86135 Augsburg, Germany
Email: martin.rasmussen@math.uni-augsburg.de

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04318-8
PII: S 0002-9947(07)04318-8
Keywords: Attractor, attractor-repeller pair, Lyapunov function, Morse decomposition, Morse set, nonautonomous dynamical system, repeller
Received by editor(s): August 2, 2005
Received by editor(s) in revised form: December 1, 2005
Published electronically: April 24, 2007
Additional Notes: This research was supported by the “Graduiertenkolleg: Nichtlineare Probleme in Analysis, Geometrie und Physik” (GK 283) financed by the DFG and the State of Bavaria
Article copyright: © Copyright 2007 American Mathematical Society