Motion in the neighborhood of a stationary point

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Abstract
A perturbed system of differential equations is considered, wherein the zero order rate of change of one variable, which depends on the other, slowly changing, variables, vanishes at a particular point. A detailed analysis of the motions in certain close neighborhoods of the stationaiy point of the perturbed system is given, the method of averaging being applied, after some preliminary transformations have been made. It is found that the motions corresponding to the averaged equations are damped under the stability conditions given in [1] by Volosov and Morgunov, who considered a more general problem. However, our analysis reveals the nature of the motion in the neighborhood of the stationary point, and, moreover, our results are valid for a time interval of the order of the reciprocal of the perturbation parameter, rather than the reciprocal of just its square root.

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