By Vladimir Igorevich Arnold (auth.)
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We shall describe a system by a point in the parameter space, that is by a point corresponding to the parameter value on the parameter axis (on the plane for two parameters, the parameter space for three and so on). We consider a division of the parameter space into two parts depending on whether the equilibrium state is stable or not. Thus we obtain on the plane (in the space) of parameters the stability domain (consisting of those values of the parameters for which the equilibrium is stable), the instability domain and dividing them the stability boundary.
In order to study the net of limit curves in the neighbourhoods of these singular * A diffeomorphism is a smooth change of variables with an inverse map that is also smooth. 52 points it is useful to consider our two-valued field of limit directions as a single-valued field on a surface of a two-fold covering of the plane ramified over K (using the terminology of complex analysis). To this end we consider the set of all directed linear elements on the plane. This set is a three dimensional manifold since the direction is determined by the point at which it is attached (two co-ordinates) and its azimuth (one angular coordinate).
At the point concerning us, 3, both families are tangent to K. From these considerations it is not difficult to see now that the net of limit curves near 3 appear as indicated in Fig. 51; above K the curves of the second family go upwards and below they go downwards (the choice of directions permits veriations which the reader can easily construct for himself). Now in Fig. 51 it is clear that to the left of 3 the accessibility boundary follows a curve of the second field of limit directions and to the right of 3 it follows a (horizontal) curve of the first field.