Controllably periodic perturbations of autonomous systems

Abstract

1. This paper is dealing with the behaviour of a periodic solution of a system of differential equations under perturbation. The case when the unperturbed system is autonomous and the perturbation periodic and non-autonomous will be considered, However, it will be assumed that the period of the perturbation is controllable. The results are stated and proved for D-periodic (derivo-periodic) solutions of cylindrical systems first (cf. [1]). Since the latter are, in a certain sense, generalizations of periodic solutions of arbitrary systems, the corresponding results for the ordinary case follow readily. The generalization involved is justified by the fact that D-periodic solutions of cylindrical systems occur frequently in applications (cf. e.g. [1], [2], [3]). It is to be noted here that already in [4] Vol. I., p. 80, H. POINCAt~ has mentioned a case which is basically the one called here a D-periodic solution of a cylindrical system.