Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 54, pp. 1-6.
Title: Liapunov exponents for higher-order linear
differential equations whose characteristic equations have
variable real roots
Author: Michael I. Gil' (Ben Gurion Univ., Israel)
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
$|x(t)|\leq \mathop{\rm const} e^{\gamma t}$ 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.