Collision and escape orbits in a generalized Hénon–Heiles model

The motion of a material point of unit mass in a field determined by a generalized Hénon–Heiles potential U=Aq12+Bq22+Cq12q2+Dq23, with (q1,q2)=(q1,q2)= standard Cartesian coordinates and (A,B,C,D)∈(0,∞)2×R2, is addressed for two limit situations: collision and escape. Using McGehee-type transformations, the corresponding collision and infinity boundary manifolds pasted on the phase space are determined. These are fictitious manifolds, but, due to the continuity with respect to initial data, their flow determines the near by orbit behaviour.The dynamics on the collision and infinity manifolds is fully described. The topology of the flow on the collision manifold is independent of the parameters. In the full phase space, while spiraling collision orbits are present, most of the orbits avoid collision. The topology of the flow on the infinity manifold changes as the ratio between CC and DD varies. More precisely, there are two symmetric pitchfork bifurcations along the line 2C−3D=02C−3D=0, due to the reshaping of the potential along the bifurcation line. Besides rectilinear and spiraling orbits, the near-escape dynamics includes oscillatory orbits, for which angular momentum alternates sign.