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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

An improved estimate for the highest Lyapunov
exponent in the method of freezing


Author: G. I. Eleutheriadis
Journal: Proc. Amer. Math. Soc. 125 (1997), 2931-2937
MSC (1991): Primary 34D08
MathSciNet review: 1403124
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Abstract: Let $\dot x=A(t)x$ and $\lambda _k(t)$ be the eigenvalues of the matrix $A(t)$. The main result of the Method of Freezing states that if $\sup _J \|A(t)\|\leq M$, $\sup _J\mathrm {max}_{1\leq k\leq n}\mathrm {Re}\,\lambda _k(t)\leq \rho$ and $\sup _J(\|A(t)-A(s)\|/|t-s|)\leq \delta$, then

\begin{displaymath}x_\mathrm {max}\leq\rho +2M\lambda _\delta ,\end{displaymath}

for the highest exponent $x_{\mathrm {max}}$ of the system, where

\begin{displaymath}\lambda _\delta =\left (\frac {C_n\delta }{4M^2}\right )^{\frac {1}{n+1}}.\end{displaymath}

The previous best known value $C_n=\frac {n(n+1)}{2}$ and the substantially smaller values of $C_n$ are reduced to the still smaller value.


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Additional Information

G. I. Eleutheriadis
Affiliation: Ektenepol, 14/3, 67100, Xanthi, Greece

DOI: https://doi.org/10.1090/S0002-9939-97-03952-X
Received by editor(s): May 1, 1996
Communicated by: Hal L. Smith
Article copyright: © Copyright 1997 American Mathematical Society