Fourier methods for quasi-periodic oscillations

Quasi-periodic oscillations and invariant tori play an important role in the study of forced or coupled oscillators. This paper presents two new numerical methods for the investigation of quasi-periodic oscillations. Both algorithms can be regarded as generalisations of the averaging and the harmonic (spectral) balance methods. The algorithms are easy to implement and require only minimal a-priory knowledge of the system. In particular, the methods do not depend on an a-priory coordinate transformation. The methods are applied to a number of illustrative examples from nonlinear electrical engineering and the results show that the methods are effcient and reliable. In addition, these examples show that the presented algorithms can also continue through regions of sub-harmonic (phase-locked) resonance even though they are designed only for the quasi-periodic case