Bifurcation Cascades and Self-similarity of Periodic Orbits with Analytical Scaling Constants in Henon-heiles Type Potentials

We investigate the isochronous bifurcations of the straight-line librating orbit in the Hénon-Heiles and related potentials. With increasing scaled energy e, they form a cascade of pitchfork bifurcations that cumulate at the critical saddlepoint 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 α and β corresponding to the two spatial directions are identical and equal to the root of the scaling constant δ that describes the geometric progression of bifurcation energies en in the limit n → ∞. The value of δ is given analytically in terms of the potential parameters.