Analytic approach to bifurcation cascades in a class of generalized Hénon-Heiles potentials

Abstract

We investigate the bifurcation cascades of a linear librational orbit in a generalized class of Henon-Heiles potentials. The stability traces of the new orbits born at its bifurcations are found numerically to intersect linearly at the saddle energy (e=1), forming what we term the ``Henon-Heiles fans''. In the limit close to the saddle energy (e-->1), where the dynamics is nearly chaotic, we derive analytical asymptotic expressions for the stability traces of both types of orbits and confirm the numerically determined properties of the generalized ``Henon-Heiles fans''. As a bonus of our results, we obtain analytical approximations for the bifurcation energies e_n which become asymptotically exact for e_n-->1.