On the bifurcations of a certain family of periodic orbits

In their numerical study of a particular family of periodic orbits in a certain potential, Contopoulos and Zikides (1980; Contopoulos, 1981) have calculated numerically successive values of the energyh at which the family becomes unstable. They showed that the ratio of successive values ofh-h _esc, whereh _esc is the energy of escape, is approximately 9.22. Here we show that the limiting ratio ash→h _esc is exp (π/√2) in this problem, and that the limiting ratio for bifurcations of this type varies from one problem to another.