  Estimates for the spectrum of the Cauchy operator and logarithms of solutions
of non-autonomous differential equations in the space, expressed in an
arbitrary matrix norm, are found. For equations with periodic coefficients, the
lower bound for the periods of "oscillatory" solutions is obtained. Similar
results are derived for a nonlinear periodic equation with zero equilibrium
position; in the autonomous case, they are valid for any periodic solution. All
the estimates are attained at a scalar equation and, thereby, are accurate for
any norm. The estimates are extended to equations with derivatives of any
order. Using the obtained results, a condition for the uniqueness of an
oscillatory solution is found; stability criteria for a linear Hamiltonian
system with periodic coefficients and for periodic solutions of a nonlinear
system (expressed through the period and any norm of the Hessian) are obtained.
Types of matrices and functions, for which the Euclidean norm has a minimum
value (and, therefore, provides the most accurate estimates for a specific
equation), are established.
