Proceedings of the American Mathematical Society

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

Author(s): G. I. Eleutheriadis
Journal: Proc. Amer. Math. Soc. 125 (1997), 2931-2937.
MSC (1991): Primary 34D08
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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

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

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

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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