Quasi-Periodic Solutions of a Damped Nonlinear Quasi-Periodic Mathieu Equation by the Incremental Harmonic Balance Method With Two Time Scales

Abstract Quasi-periodic (QP) solutions of a damped nonlinear QP Mathieu equation with cubic nonlinearity are investigated by using the incremental harmonic balance (IHB) method two time scales. The contains incommensurate excitation frequencies, one is small frequency while other nearly equals twice linear natural frequency. It found that Fourier spectra consist uniformly spaced sidebands due to nonlinearity. IHB scales, which relates adopted trace solution curves in an automatical way and find all frequencies their corresponding amplitudes. Effects parametric studied detail. Based on approximation periodic large period, Floquet theory used study stability solutions. Three types can be obtained from method, agree very well results numerical integration. However, perturbation double-step multiple scales (MMS) obtains only type since ratio first reduced-modulation 1 second procedure, do not need ratio. Furthermore, MMS different those integration