Parametric resonance as a model for QPO sources -- I. A general approach to multiple scales

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 a quadratic nonlinearity and we show that this case is particularly relevant for QPOs. We present a very convenient way how to study internal resonances of a fully general system using the method of multiple scales. Finally, we concentrate to conservative systems and discuss their behavior near the 3:2 resonance.