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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
DOI: https://doi.org/10.1090/S0002-9939-97-03952-X
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 $\sup _J \|A(t)\|\leq M$, $\sup _J\max _{1\leq k\leq n}\mathrm {Re} \lambda _k(t)\leq \rho$ and $\sup _J(\|A(t)-A(s)\|/|t-s|)\leq \delta$, then \[ x_{\max }\leq \rho +2M\lambda _\delta ,\] for the highest exponent $x_{\max }$ of the system, where \[ \lambda _\delta =\left (\frac {C_n\delta }{4M^2}\right )^{\frac {1}{n+1}}.\] 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

Received by editor(s): May 1, 1996
Communicated by: Hal L. Smith
Article copyright: © Copyright 1997 American Mathematical Society