On Linear Perturbations Decaying at Infinityand Altering the Lyapunov Exponentsof Regular Linear Differential Systems

We prove that for any real-valued function $$\theta (\cdot ) $$ defined on the half-line $$[0,+\infty ) $$ , monotone increasing to $$+\infty $$ , and such that $$\theta (t)/t\to 0 $$ as $$t\to {+\infty }$$ and for any positive integer $$n\geq 2$$ there exists an $$n $$ -dimensional Lyapunov regular linear differential system whose Lyapunov exponents change under some perturbation of its coefficient matrix with the norm of at most $$\mathrm {const}\exp \{-\theta (t)\} $$ for all $$t\geq 0 $$ . Previously, examples of such regular systems were known only for $$ \theta (t)=\sqrt {t}$$ .