Limit-Cycles and Rotated Vector Fields

The problem of limit-cycles in the plane was formulated by Poincar6 (7), who pointed out the existence and importance of these special solutions. The question at issue is to find a systematic method for determining the existence and location of the cycles of a given second-order system of ordinary differential equations. The importance of the problem lies in the fact that its solution would enable us to determine the limiting sets of non-periodic motions. This paper contains a contribution to the solution of the general problem, based on a method of complete families of rotated vector fields. In this method the vector field is varied so that the qualitative properties change in a controlled manner. The theory of complete families is first developed, and the limitcycle problem is then treated as an application. For this purpose the cycles of a given vector field are classified according to the critical points they enclose. In the simplest case, namely when the cycles in question enclose only one critical point, which is elementary, a general test is given by which the existence and location problem can be solved. More general cases can be solved if a Bendixson curve enclosing the critical points in question can be found. The results of this paper were submitted in the author's doctoral thesis, Princeton 1951. Acknowledgement is due the Faculty of Princeton University for the Proctor Fellowship held at that time. The author wishes also to thank Professors S. Lefschetz, M. Schiffer, and S. Diliberto for guidance and advice.