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Oscillatory instabilities in Hamiltonian anharmonic lattices are known toappear through Hamiltonian Hopf bifurcations of certain time-periodic solutionsof multibreather type. Here, we analyze the basic mechanisms for this scenarioby considering the simplest possible model system of this kind where theyappear: the three-site discrete nonlinear Schr\"odinger model with periodicboundary conditions. The stationary solution having equal amplitude andopposite phases on two sites and zero amplitude on the third is known to beunstable for an interval of intermediate amplitudes. We numerically analyze thenature of the two bifurcations leading to this instability and find them to beof two different types. Close to the lower-amplitude threshold stabletwo-frequency quasiperiodic solutions exist surrounding the unstable stationarysolution, and the dynamics remains trapped around the latter so that inparticular the amplitude of the originally unexcited site remains small. Bycontrast, close to the higher-amplitude threshold all two-frequencyquasiperiodic solutions are detached from the unstable stationary solution, andthe resulting dynamics is of 'population-inversion' type involving also theoriginally unexcited site.Comment: 25 pages, 11 figures, to be published in J. Phys. A: Math. Gen.  Revised and shortened version with few clarifying remarks adde

journal of physics. a, mathematical and general/journal of physics. a, mathematical and general

Hamiltonian Hopf bifurcations in the discrete nonlinear Schrödinger trimer: oscillatory instabilities, quasi-periodic solutions and a new type of self-trapping transition