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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

- V. M. Alekseev,
*The asymptotic behaviour of solutions of slightly non-linear systems of ordinary differential equations*, Soviet Math. Dokl.**1**(1960), 1035–1038. MR**0114983** - V. M. Alekseev and R. È. Vinograd,
*On the method of “freezing”*, Vestnik Moskov. Univ. Ser. I Mat. Meh.**21**(1966), no. 5, 30–35 (Russian, with English summary). MR**0200572** - B. F. Bylov, R. È. Vinograd, D. M. Grobman, and V. V. Nemyckiĭ,
*Teoriya pokazateleĭ Lyapunova i ee prilozheniya k voprosam ustoĭ chivosti*, Izdat. “Nauka”, Moscow, 1966 (Russian). MR**0206415** - Ju. L. Dalec′kiĭ and M. G. Kreĭn,
*Stability of solutions of differential equations in Banach space*, American Mathematical Society, Providence, R.I., 1974. Translated from the Russian by S. Smith; Translations of Mathematical Monographs, Vol. 43. MR**0352639** - Ju. I. Elefteriadi,
*Attainability of the estimate of the highest exponent in the freezing method in the case $n>2$*, Differencial′nye Uravnenija**10**(1974), 1379–1386, 1583 (Russian). MR**0361299** - I. M. Glazman and Yu. I. Lyubich,
*Konechnomernyĭ lineĭ nyĭ analiz v zadachakh*, Izdat. “Nauka”, Moscow, 1969 (Russian). MR**0354715** - N. A. Izobov,
*Cases of sharpening and the attainability of an estimate of the highest exponent in the method of freezing*, Differencial′nye Uravnenija**7**(1971), 1179–1191, 1339–1340 (Russian). MR**0293197** - N. A. Izobov,
*Refinement of bounds for highest exponents in the freezing method*, Differentsial′nye Uravneniya**19**(1983), no. 8, 1454–1456 (Russian). MR**714814** - N. Ya. Lyǎšcenko,
*On the asymptotic stability of the solutions of system differential equations*, Dokl. Akad. Nauk SSSR,**96**(1954), no. 2. (Russian) - Robert E. Vinograd,
*An improved estimate in the method of freezing*, Proc. Amer. Math. Soc.**89**(1983), no. 1, 125–129. MR**706524**, DOI https://doi.org/10.1090/S0002-9939-1983-0706524-1 - J. H. Wilkinson,
*The algebraic eigenvalue problem*, Clarendon Press, Oxford, 1965. MR**0184422** - Reginald P. Tewarson,
*Sparse matrices*, Academic Press, New York-London, 1973. Mathematics in Science and Engineering, Vol.99. MR**0362875**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
34D08

Retrieve articles in all journals with MSC (1991): 34D08

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