On periodic solutions of nonlinear autonomous differential systems in the critical case

Abstract

We propose a new numerical-analytic algorithm for the investigation of periodic solutions of nonlinear autonomous systems of ordinary differential equations in the critical case.The present paper is devoted to the investigation of the existence of periodic solutions of a nonlinear autonomous system of differential equations with separated linear part, namelyin the case where all solutions of the corresponding linear differential system have a certain common period T, and the period of a solution of the nonlinear system either coincides with T or is close to it. Analogous problems were considered in numerous works (see, e.g., [1][2][3][4][5]). For the solution of this problem, one uses a modification of the numerical-analytic method [6,7], the specific feature of which lies in the fact that the restrictions imposed on the Lipschitz matrix are applied not to the entire right-hand side but only to the nonlinearity g(x). T -Periodic Solutions of a Nonlinear SystemWe consider the nonlinear autonomous system of ordinary differential equations (1) in which J is an n × n real constant matrix such that the corresponding linear homogeneous systemhas n nontrivial solutions of period T = 2Π/ν. We investigate the problem of the existence and construction of periodic solutions of system (1) whose period coincides with the period T of solutions of the corresponding linear system (2). It is clear that the eigenvalues of the matrix J are either zero or purely imaginary. Since this matrix can always be reduced to a canonical skew-symmetric matrix by a similarity transformation, we can assume, without loss of generality, thatAssume that, in the domain