Bifurcation cascades and self-similarity of periodic orbits with analytical scaling constants in Henon-Heiles type potentials

Abstract

We investigate the isochronous bifurcations of the straight-line librating orbit in the Henon-Heiles and related potentials. With increasing scaled energy e, they form a cascade of pitchfork bifurcations that cumulate at the critical saddle-point energy e=1. The stable and unstable orbits created at these bifurcations appear in two sequences whose self-similar properties possess an analytical scaling behavior. Different from the standard Feigenbaum scenario in area preserving two-dimensional maps, here the scaling constants \alpha and \beta corresponding to the two spatial directions are identical and equal to the root of the scaling constant \delta that describes the geometric progression of bifurcation energies e_n in the limit n --> infinity. The value of \delta is given analytically in terms of the potential parameters.