Analysis of the Existence of Periodic Solutions of the Systems of Nonlinear Differential Equations with a Small Parameter

We consider the existence of periodic solutions-cycles in nonlinear differential eq\-u\-a\-ti\-ons with a small parameter. We obtain necessary and sufficient conditions for the existence of periodic solutions. These conditions significantly expand the applicability of the Pontryagin small parameter method from the theory of dynamical systems on the plane. We do not assume the differentiability of all functions involved in the system. Moreover, the system is not Hamiltonian. In order to prove the existence of periodic solutions of the system of nonlinear differential equations we use topological methods of non\-li\-ne\-ar analysis. Based on the proposed methods, we formulate and prove theorems on the necessary and sufficient conditions for the existence of periodic solutions under the condition of continuity of all functions involved in the system. Moreover, we use the transition to the polar coordinate system and Jordan transformations. In the last section we propose a method for developing examples for a specific class of functions. Furthermore, we give an example of a system such that we easy verify the conditions for the existence of periodic solutions for small values of $\varepsilon$.