Stability of an integral curve y(x) of ODE at a singular set of the second type

Abstract

A second order ODE (1) can have singular set S of the second type (see Introduction). We study the stability of behaviour of an integral curve y(x) at S with respect to passage from (1) to perturbed equation (2). Before such stability was studied only for “inverse” integral curve x(y) (see, for example, [1–4] and some other our publications). In particular, all the local extremas of the both integral curves belong to the union of the singular sets with the cartesian axis Y : y = 0. Much of properties of y(x) cannot be obtained as consequence of the properties of x(y) and vice-versa. As an example of application of obtained results we study the well known Van der Pol’s equation indicating some its properties not explained before.