Non-Universality for a Class of Bifurcations

Abstract

Contopoulos and Zikides have investigated numerically a family of periodic orbits (in a certain potential) which passes from stability to instability infinitely often, at values of the energy, h_k having a finite limit point, h_esc, as k → ∞. From eight such transitions they found numerically that the ratio $\frac{{{h_k} - {h_{esc}}}}{{{h_{k + 1}} - {h_{esc}}}}$ tends to a number close to 9.22.

In this paper it is shown analytically that the limiting ratio is exp $(\pi /\sqrt 2 )$. Furthermore, it is shown that the limiting ratio varies from one problem to another, and problems can be constructed in which it takes any value above 1. Thus there is no “quantitative universality” such as has been found for other types of infinite bifurcation by Feigenbaum and others.