Internal resonance in non-linear disk oscillations and the amplitude evolution of neutron star kilohertz QPOs

We explore some properties of twin kilohertz quasiperiodic oscillations (QPOs) in a simple toy-model consisting of two oscillation modes coupled by a general nonlinear force. We examine resonant effects by slowly varying the values of the tunable, and nearly commensurable, eigenfrequencies. The behavior of the actual oscillation frequencies and amplitudes during a slow transition through the 3:2 resonance is examined in detail and it is shown that both are significantly affected by the nonlinearities in the governing equations. In particular, the amplitudes of oscillations reflect a resonant exchange of energy between the modes, as a result the initially weaker mode may become dominant after the transition. We note that a qualitatively similar behavior has been recently reported in several neutron star sources by Torok (2008, arXiv:0812.4751), who found that the difference of amplitudes in neutron star twin peak QPOs changes sign as the observed frequency ratio of the QPOs passes through the value 3:2.