# Uniform Stability of Local Extrema of an Integral Curve of an ODE of Second Order

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## Abstract

A second-order equation can have singular sets of first and second type, S_{1} and S_{2} (see the introduction), where the integral curve *x(y)* does not exist in the ordinary sense but where it can be extended by using the first integral [1–-5]. Denote by *Y* the Cartesian axis *y*=0. If the function *x(y)* has a derivative at a point of local extremum of this function, then this point belongs to *S*_{1}∪*Y*. The extrema at which *y'(x)* does not exist can be placed on *S*_{2}. In [5–-8], the stability and instability of extrema on *S*_{1}∪ *S*_{2} under small perturbations of the equation were considered, and the stability of the mutual arrangement of the maxima and minima of *x*(*y*) on the singular set was studied (locally as a rule, i.e., in small neighborhoods of singular points). In the present paper, sufficient conditions for the preservation of type of a local extremum on the finite part of *S*_{1} or *S*_{2} are found for the case in which the perturbation on all of this part does not exceed some explicitly indicated quantity which is the same on the entire singular set.

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