Jorke's Theorem and Wirtinger's Inequality

We consider autonomous systems of ordinary differential equations (of first or higher order) whose right-hand sides satisfy the Lipschitz condition stated in terms of the Euclidean metric and of nonnegative matrices. Using Wirtinger's inequality, we prove theorems on the lower bounds for the periods of periodic nonstationary solutions of autonomous systems, which generalize Jorke's theorem. In the case of nonnegative indecomposable matrices we discuss the sharpness of the estimates obtained.