Lower bounds for the maximal Lyapunov exponent in one-parameter families of linear differential systems

We consider a two-dimensional, nonautonomous, homogeneous system of linear ordinary differential equations that depends on the real parameter $\mu$. It is assumed that the Cauchy operator for each unit time interval is a product of a rotation matrix by an angle, the value of which is an affine function of $\mu$, and of a diagonal matrix with a unit determinant, which is chosen to be close to a constant and whose norm is sufficiently large to guarantee the monotonicity with respect to $\mu$ of polar angle for any solution to the system. This class of systems contains an example of a non-almost-reducible linear system with limit-periodic coefficients constructed by V. M. Millionshchikov. We use his rotation method to establish the positivity of the maximal Lyapunov exponent in one-parameter family for some set of parameter values that has positive Lebesgue measure. To derive this result, we prove the monotonicity with respect to $\mu$ of angles in singular-value decomposition for Cauchy operator and moreover that its derivative is separated from zero. Further, the angle itself increases as a monotonic linear function of $t$. Both of these properties, by induction, give us a small average loss for the Cauchy operator norm on exponentially growing time intervals, which leads to its exponential growth as a function of $t$.