  The time-dependent extended Gross-Pitaevskii equation for Bose-Einstein
condensates with attractive 1/r interaction is investigated with both a
variational approach and numerically exact calculations. We show that these
condensates exhibit signatures known from the nonlinear dynamics of autonomous
Hamiltonian systems. The two stationary solutions created in a tangent
bifurcation at a critical value of the scattering length are identified as
elliptical and hyperbolical fixed points, corresponding to stable and unstable
stationary states of the condensate. The stable stationary state is surrounded
by elliptical islands, corresponding to condensates periodically oscillating in
time, whereas condensate wave functions in the unstable region undergo a
collapse within finite time. For negative scattering lengths below the tangent
bifurcation no stationary solutions exist, i.e., the condensate is always
unstable and collapses.
