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Dilute ultracold atom clouds in the Bose-Einstein condensed phase with long-range atomic interactions, in addition to the contact interaction, open the possibility of studying the properties of degenerate quantum gases in which the relative strengths of long- and short-range interactions can be adjusted continuously by tuning the contact interaction via a Feshbach resonance. Examples are condensates of neutral atoms with electromagnetically induced attractive 1/r interaction, and condensates of atoms with a large magnetic dipole moment, such as chromium. In both types of condensates universal stability thresholds exist where collapse of the condensates sets in. I show that these thresholds in fact correspond to bifurcation points where always two solutions of the Gross-Pitaevskii equation disappear in a tangent bifurcation, one dynamically stable and the other unstable. I point out that the thresholds correspond to "exceptional points", i.e. branching singularities of the Hamiltonian. The dynamics of excited condensate wave functions can be analyzed via Poincare surfaces of section, and shows both regular and chaotic motion, corresponding to (quasi-) periodically oscillating and irregularly fluctuating condensates, respectively. |