A Study of Trajectories which Tend to a Limit Cycle in These-Space

Abstract

The object of this paper is to study the behavior of a solution curve of an ordinary differential equation as this solution curve approaches a limit cycle of the differential equation in Euclidian 3-space. In the entire paper we shall denote the limit cycle by the symbol C and the solution curve which tends to C by the symbol S. The same problem if studied in 2-space has a very simple solution. A solution curve which approaches a limit cycle in the plane spirals towards this limit cycle from one of the two sides of the limit cycle. The manner in which a trajectory can approach a limit cycle in 3-space was studied by Birkhoff [1]. But in his classical paper on transformations of surfaces Birkhoff assumes conservation of energy. No such assumption will be made here. The methods of this paper are similar to the methods used by one of us (2] in investigating singular points in 3-space. We shall use some of the results obtained there. In Section 2 the exact problem is stated and certain transformations of coordinates carried out which facilitate the analysis. In Section 3 two tori, T, and T2, are constructed. T1 is essentially the torus of radius e with the limit cycle as center line. The other torus T2 is produced roughly by expanding C in another 3-space. If i is the identity map from the second 3-space onto the first one it is then shown that a certain set on T, which is related to the limiting behavior of S is just the image under i of a set on T2 about which it is fairly simple to get information. Section 4 is devoted to the study of limit sets on T2. Finally, in Section 5 the main theorem is proved which shows how a solution curve can behave if it approaches a limit cycle.