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The Lyapunov spectrum of families of time-varying matrices

Authors: Fritz Colonius and Wolfgang Kliemann
Journal: Trans. Amer. Math. Soc. 348 (1996), 4389-4408
MSC (1991): Primary 34D08, 93B05, 58F25
MathSciNet review: 1329531
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Abstract: For $L^{\infty }$-families of time varying matrices centered at an unperturbed matrix, the Lyapunov spectrum contains the Floquet spectrum obtained by considering periodically varying piecewise constant matrices. On the other hand, it is contained in the Morse spectrum of an associated flow on a vector bundle. A closer analysis of the Floquet spectrum based on geometric control theory in projective space and, in particular, on control sets, is performed. Introducing a real parameter $\rho \ge 0$, which indicates the size of the $L^{\infty }$-perturbation, we study when the Floquet spectrum, the Morse spectrum, and hence the Lyapunov spectrum all coincide. This holds, if an inner pair condition is satisfied, for all up to at most countably many $\rho $-values.

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Additional Information

Fritz Colonius
Affiliation: Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany

Wolfgang Kliemann
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011

Keywords: Lyapunov exponents, Floquet exponents, Lyapunov regularity, Morse spectrum, control sets, chain control sets, chain recurrence
Received by editor(s): January 25, 1994
Received by editor(s) in revised form: March 31, 1995
Additional Notes: This research was partially supported by DFG grant no. Co 124/8-2 and ONR grant no. N00014-93-1-0868
Article copyright: © Copyright 1996 American Mathematical Society