The autoparametric 3:2 resonance in conservative systems

In the resonance model, high-frequency quasi-periodic oscillations (QPOs) are supposed to be a consequence of nonlinear resonance between modes of oscillations occurring within the innermost parts of an accretion disk. Several models with a prescribed mode--mode interaction were proposed in order to explain the characteristic properties of the resonance in QPO sources. In this paper, we examine nonlinear oscillations of a system having two degrees of freedom and we show that this case could be particularly relevant for QPOs. We present a very convenient way how to study autoparametric resonances of a fully general system using the method of multiple scales. We concentrate to conservative systems and discuss their behavior near the 3:2 parametric resonance.