Bounds for periodic solutions of vector second-order nonlinear differential equations

Upper bounds for the amplitudes and lower bounds for the periods of periodic solutions of coupled second-order nonlinear differential equations are found. The results are expressed in a solution of a positive operator equation which is associated with the considered system. Cases are indicated for which the obtained bounds are precise and surprising behavior of the amplitude and periods of such systems under insertion of certain terms is revealed. The results are illustrated by examples.