Investigation of the periodic solutions of nonlinear autonomous systems in the critical case

Abstract

517.925We analyze the conditions of existence and the numerical-analytic method for the approximate construction of periodic solutions of nonlinear autonomous systems of differential equations in the critical case.The theory of periodic boundary-value problems is extensively used in the investigation of various engineering and natural processes. For this reason, significant attention of numerous researchers is given to the problems of existence and approximate construction of periodic solutions of differential equations (see, e.g., [1 -4]). Note that an important place in this field of investigations is occupied by the analysis of autonomous systems of differential equations [5 -7].In the present paper, we continue our investigations originated in [8,9] and propose a modification of the Samoilenko numerical-analytic method aimed at the investigation of existence and approximate construction of periodic solutions of nonlinear autonomous systemsin the critical case. In this case, the restrictions imposed on the Lipschitz matrix are valid not for the entire righthand side but only for the nonlinearity.