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Estimates for the spectrum of the Cauchy operator and logarithms of solutionsof non-autonomous differential equations in the space, expressed in anarbitrary matrix norm, are found. For equations with periodic coefficients, thelower bound for the periods of "oscillatory" solutions is obtained. Similarresults are derived for a nonlinear periodic equation with zero equilibriumposition; in the autonomous case, they are valid for any periodic solution. Allthe estimates are attained at a scalar equation and, thereby, are accurate forany norm. The estimates are extended to equations with derivatives of anyorder. Using the obtained results, a condition for the uniqueness of anoscillatory solution is found; stability criteria for a linear Hamiltoniansystem with periodic coefficients and for periodic solutions of a nonlinearsystem (expressed through the period and any norm of the Hessian) are obtained.Types of matrices and functions, for which the Euclidean norm has a minimumvalue (and, therefore, provides the most accurate estimates for a specificequation), are established.

Sharp estimates for spectrum of Cauchy operator and periods of solutions of Lipschitz differential equations