Lower bounds for the upper Lyapunov exponent in one-parameter families of Millionshchikov systems

For a special class of two-dimensional differential systems of the form $\dot {x}=(A(t)+\mu B(t))x$, in particular, the Lyapunov improper almost periodic systems constructed by V. M. Millionshchikov, it is shown that the upper characteristic exponent $\lambda_2(A + \mu B)$ is positive for all $\mu$ from a set of positiv Lebesgue measure.