A novel approach for generating multi-direction multi-double-scroll attractors

Attention is focused in this work on quasiperiodic motion of nonlinear systems whose spectrum contains uniformly spaced sideband frequencies with a distance \(\omega _{d}\) apart, around a frequency \(\omega \) with \(\omega \gg \omega _{d}\) and its integer multiples, which are referred to as carrier frequencies. The ratio of the two frequencies \(\omega \) and \(\omega _{d}\) is an irrational number. A new method based on the traditional incremental harmonic balance (IHB) method with multiple timescales, referred to as Lau method, where two timescales, \(\tau _{1}=\omega t\) (a fast timescale) and \(\tau _{2}=\omega _{d}t\) (a slow timescale), are introduced, is presented to analyze quasiperiodic motion of nonlinear systems. An amplitude increment algorithm is adapted to deal with cases where the two frequencies \(\omega \) and \(\omega _{d}\) are    unknown a priori, in order to automatically trace frequency response of quasiperiodic motion of nonlinear systems and accurately calculate all frequency components and their corresponding amplitudes. Results of application of the present IHB method to quasiperiodic free vibration of a hinged–clamped beam with internal resonance between two transverse modes are shown and compared with previously published results with Lau method and those from numerical integration. While differences are noted between results predicted by the present IHB method and Lau method, excellent agreement is achieved between results from the present IHB method and numerical integration even in cases of strongly nonlinear vibration. The present IHB method is also used to analyze quasiperiodic free vibration of high-dimensional models of the hinged–clamped beam.