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

We investigate the bifurcation cascades of a linear librational orbit in a generalized class of Hénon-Heiles potentials. The stability traces of the new orbits created at its bifurcations are found numerically to intersect linearly at the saddle energy (e=1) , forming what we term the "Hénon-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 Hénon-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 .