1. This paper presents a bifurcation theorem for the second order system /l(Xi, x2, p), f2 (xi, x2, p). A description of the classical bifurcation problem will be found in Minorsky [l, Chapter V] and Andronov and Chaikin [2, Chapter 6]. Here the functions /i and /2 are required to be analytic and the proof depends on the series expansion guaranteed by the analyticity. In [3, Chapter IV, §6], K. 0. Friedrichs establishes a bifurcation theorem which does not require analyticity but uses the

more »

... it function theorem. Continuous second derivatives are required. Since the differentiability conditions are less severe, the theorem presented here could apply when the theorem of Friedrichs does not. However, the purpose of the paper is to show the bifurcation process geometrically-to show that it arises from the properties of trajectories considered as point sets and from the continuity of the vector field. To this end, the existence of a certain dynamical system will be postulated. This is in contrast to the above mentioned bifurcation theorems where conditions are prescribed (on certain coefficients, matrices, etc.) to guarantee the existence of an appropriate system. This will show the dependence of the theorem on the structure of the dynamical system rather than on analytical conditions necessary to achieve it. (pi, p2) in the plane is said to be a stable (unstable) spiral point if there is a region R such that the trajectory (oo (t->-oo) and (ii) that the function w(t)= tan'1 (2(t)/i(t)) is such that \w(t)\->°o as t->°o(t-^>-oo) [4, p. 376]. It will be assumed that the region R above has been extended to a canonical region in the sense of Markus [5] . The solution curves which bound these regions are called séparatrices. For a precise defini- A critical point p=