Electron. J. Diff. Eqns., Vol. 2008(2008), No. 54, pp. 1-6.

Liapunov exponents for higher-order linear differential equations whose characteristic equations have variable real roots

Michael I. Gil'

Abstract:
We consider the linear differential equation
$$ \sum_{k=0}^n a_k(t)x^{(n-k)}(t)=0\quad t\geq 0, \; n\geq 2, $$
where $a_0(t)\equiv 1$, $a_k(t)$ are continuous bounded functions. Assuming that all the roots of the polynomial $z^n+a_1(t)z^{n-1}+ \dots +a_n(t)$ are real and satisfy the inequality $r_k(t)<\gamma$ for $t\geq 0$ and $k=1, \dots, n$, we prove that the solutions of the above equation satisfy $|x(t)|\leq \hbox{ const} e^{\gamma t}$ for $t\geq 0$.

Submitted December 27, 2007. Published April 15, 2008.
Math Subject Classifications: 34A30, 34D20.
Key Words: Linear differential equations; Liapunov exponents; exponential stability.

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Michael I. Gil'
Department of Mathematics
Ben Gurion University of the Negev
P.0. Box 653, Beer-Sheva 84105, Israel
Email: gilmi@cs.bgu.ac.il

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