Morse decompositions of nonautonomous dynamical systems

Rasmussen, Martin

doi:10.1090/S0002-9947-07-04318-8

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
DOI: https://doi.org/10.1090/S0002-9947-07-04318-8
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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