On a priori estimate and the existence of periodic solutions for a class of systems of nonlinear ordinary differential equations

In this paper is investigated the question of an a priori estimate and the existence of periodic solutions for one class of systems of ordinary differential equations, in which the main nonlinear part is gradient of a positively homogeneous function. Found the necessary and sufficient conditions that provide an a priori estimate for periodic solutions. It is proved that under the conditions of an a priori estimate, periodic solutions exist if and only if not equal to zero degree mapping of the gradient of a positively homogeneous function on the unit sphere. The novelty of the work is that, firstly, previously obtained results of the authors are generalized for multidimensional systems, and secondly, is proved the formula for calculating degree mapping of the gradient of a positively homogeneous function on the unit sphere.