  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.
