Bifurcations in biparametric quadratic potentials.

Numerous dynamical systems are represented by quadratic Hamiltonians with the phase space on the S (2) sphere. For a class of these Hamiltonians depending on two parameters, we analyze the occurrence of bifurcations and we obtain the bifurcation lines in the parameter plane. As the parameters evolve, the appearance-disappearance of homoclinic orbits in the phase portrait is governed by three types of bifurcations, the pitchfork, the teardrop and the oyster bifurcations. We find that the teardrop bifurcation is associated with a non-elementary fixed point whose Poincare index is zero. (c) 1995 American Institute of Physics.