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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Morse decompositions of nonautonomous dynamical systems
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by Martin Rasmussen PDF
Trans. Amer. Math. Soc. 359 (2007), 5091-5115 Request permission


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
  • MR Author ID: 751819
  • Email:
  • 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
  • © Copyright 2007 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 359 (2007), 5091-5115
  • MSC (2000): Primary 34D05, 37B25, 37B55, 37C70; Secondary 34D08
  • DOI:
  • MathSciNet review: 2320661