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

Gil', Michael I.

Electronic Journal of Differential Equations, Tome 2008 (2008).

Résumé

Summary: 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 $t\geq 0$.

Détail

Classification : 34A30, 34D20
Keywords: linear differential equations, Liapunov exponents, exponential stability