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All-time Morse decompositions of linear nonautonomous dynamical systems


Author: Martin Rasmussen
Journal: Proc. Amer. Math. Soc. 136 (2008), 1045-1055
MSC (2000): Primary 34D05, 37B25, 37B55, 37C70, 39A11
DOI: https://doi.org/10.1090/S0002-9939-07-09071-5
Published electronically: November 28, 2007
MathSciNet review: 2361880
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Abstract | References | Similar Articles | Additional Information

Abstract: Morse decompositions provide inside information about the global asymptotic behavior of dynamical systems on compact metric spaces. Recently, the existence of Morse decompositions for nonautonomous dynamical systems was proved by restricting attention to the past or the future of the system, but in general, such a construction is not realizable for the entire time. In this article, it is shown that all-time Morse decompositions can be defined for linear systems on the projective space. Moreover, the dynamical properties are discussed and an analogue to the Theorem of Selgrade is proved.


References [Enhancements On Off] (What's this?)

  • 1. L. Arnold, Random Dynamical Systems, Springer, Berlin Heidelberg New York, 1998. MR 1723992 (2000m:37087)
  • 2. F. Colonius and W. Kliemann, The Dynamics of Control, Birkhäuser, 2000. MR 1752730 (2001e:93001)
  • 3. C. C. Conley, Isolated Invariant Sets and the Morse Index, Regional Conference Series in Mathematics, no. 38, American Mathematical Society, Providence, Rhode Island, 1978. MR 511133 (80c:58009)
  • 4. W. A. Coppel, Dichotomies in Stability Theory, Springer Lecture Notes in Mathematics, vol. 629, Springer, Berlin Heidelberg New York, 1978. MR 0481196 (58:1332)
  • 5. K. Palmer and S. Siegmund, Generalized Attractor-Repeller Pairs, Diagonalizability and Integral Separation, Advanced Nonlinear Studies 4 (2004), 189-207. MR 2060649 (2005b:37041)
  • 6. M. Rasmussen, Attractivity and Bifurcation for Nonautonomous Dynamical Systems, Lecture Notes in Mathematics 1907, Springer, 2007.
  • 7. M. Rasmussen, Morse Decompositions of Nonautonomous Dynamical Systems, Transactions of the American Mathematical Society 359 (2007), 5091-5115.
  • 8. M. Rasmussen, Dichotomy Spectra and Morse Decompositions of Linear Nonautonomous Differential Equations, submitted.
  • 9. R. J. Sacker and G. R. Sell, A Spectral Theory for Linear Differential Systems, Journal of Differential Equations 27 (1978), 320-358. MR 0501182 (58:18604)
  • 10. J. F. Selgrade, Isolated Invariant Sets for Flows on Vector Bundles, Transactions of the American Mathematical Society 203 (1975), 359-390. MR 0368080 (51:4322)
  • 11. G. R. Sell, Nonautonomous Differential Equations and Dynamical Systems, Transactions of the American Mathematical Society 127 (1967), 241-283. MR 0212313 (35:3187a)
  • 12. G. R. Sell, Topological Dynamics and Ordinary Differential Equations, Van Nostrand Reinhold Mathematical Studies, London, 1971. MR 0442908 (56:1283)
  • 13. S. Siegmund, Dichotomy Spectrum for Nonautonomous Differential Equations, Journal of Dynamics and Differential Equations 14 (2002), no. 1, 243-258. MR 1878650 (2002j:34082)

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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: https://doi.org/10.1090/S0002-9939-07-09071-5
Keywords: Attractor, attractor-repeller pair, Morse decomposition, Morse set, nonautonomous dynamical system, projective space, repeller
Received by editor(s): July 11, 2006
Received by editor(s) in revised form: January 16, 2007
Published electronically: November 28, 2007
Additional Notes: Research supported by Bayerisches Eliteförderungsgesetz of the State of Bavaria, Germany
Communicated by: Jane M. Hawkins
Article copyright: © Copyright 2007 American Mathematical Society

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