Twin-peak quasiperiodic oscillations as an internal resonance

Two inter-related peaks occur in high-frequency power spectra of X-ray lightcurves of several black-hole candidates. We further explore the idea that a non-linear resonance mechanism, operating in strong-gravity regime, is responsible for these quasi-periodic oscillations (QPOs). By extending the multiple-scales analysis of Rebusco, we construct two-dimensional phase-space sections, which enable us to identify different topologies governing the system and to follow evolutionary tracks of the twin peaks. This suggests that the original (Abramowicz and Kluzniak) parametric-resonance scheme can be viewed as an ingenuous account of the QPOs model with an internal resonance. We show an example of internal resonance in a system with up to two critical points, and we describe a general technique that permits to treat other cases in a systematical manner. A separatrix divides the phase-space sections into regions of different topology: inside the libration region the evolutionary tracks bring the observed twin-peak frequencies to an exact rational ratio, whereas in the circulation region the observed frequencies remain off resonance. Our scheme predicts the power should cyclically be exchanged between the two oscillations. Likewise the high-frequency QPOs in neutron-star binaries, also in black-hole sources one expects, as a general property of the non-linear model, that slight detuning pushes the twin-peak frequencies out of sharp resonance.