On a Priori Estimate and Existence of Periodic Solutions for a Class of Systems of Nonlinear Ordinary Differential Equations

A question on a priori estimate and existence of periodic solutions for one class of systems of ordinary differential equations, where the principal nonlinear part is singled out, which is a gradient of a positively homogeneous function, is studied. Necessary and sufficient conditions on the coefficients of the positively homogeneous function are found such that the a priori estimate of periodic solutions holds. Given the a priori estimate, the periodic solutions are proved to exist if and only if the rotation of the gradient of the positively homogeneous function is nonzero on the unit sphere. The novelty of the work is that, firstly, the authors' earlier results are generalized to include multidimensional systems and, secondly, the formula for calculating the rotation of the gradient of the positively homogeneous function on the unit sphere is proved.