Limit Cycles of Systems of the Second Order.

The importance in non-linear mechanics of these limit cycles, or isolated periodic solutions, is well known. It has led to a variety of methods for finding these solutions which yield some information in noteworthy special cases. In Princeton lectures in the fall of 1949, Lefschetz introduced and applied to certain questions the following method: a rotation by a fixed angle a being applied to the vector field (X, Y) in the phase plane of the system (1) gave rise to a new vector field (X cos a Y sin a, X sin a + Y cos a), whose trajectories were then compared to those of the original system. A method essentially like that of Lefschetz is utilized in this note, and has made it possible to answer systematically many questions concerning limit cycles of second order systems.Consider then a family of systems = P(x, y, a), y = Q(x, y, a). (2)