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

TL;DR

This paper introduces a general multiple scales approach to analyze nonlinear oscillations with quadratic nonlinearity, focusing on internal resonances like the 3:2 ratio relevant for high-frequency QPOs in accretion disks.

Contribution

It develops a versatile method using multiple scales to study nonlinear oscillations and internal resonances, especially near the 3:2 resonance, applicable to QPO models.

Findings

The method effectively analyzes internal resonances in nonlinear systems.

Quadratic nonlinearity is particularly relevant for modeling QPOs.

Behavior near the 3:2 resonance is characterized for conservative systems.

Abstract

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.