Continuation of quasi-periodic solutions with two-frequency harmonic balance method

International audience; Nonlinear systems can have periodic solutions evolving with the parameters of the system. Studying this evolution (numerical continuation of solutions) uncovers sought-after regimes in musical acoustics : many musical instruments rely on auto-oscillation, that is, the excitation of a nonlinear system coupled with a linear resonator, where some parameters may be adjusted by the player. Periodic solutions can be approximated as truncated Fourier series (Harmonic Balance Method) ; the period is one of the unknowns. Several stable or unstable solutions can be found for the same playing parameters thanks to continuation. An important challenge is the continuation of quasi-periodic solutions, also called multiphonic sounds by musicians. Depending on the context, these oscillation regimes are considered pleasant (jazz or contemporary music for instance) or unpleasant (classical music). We developed a method based on double Fourier series, coupled with a continuation technique. The two base frequencies are unknowns and incommensurable. The system is reformulated as quadratic in order to allow straight interface with previous work on periodic harmonic balance. This method is illustrated on simple models relevant to musical acoustics, though the method can be applied to many nonlinear problems, without a priori knowledge of the solutions.