International audienceCombinig the harmonic balance method (HBM) and a continuation method is a well-known technique to follow the periodic solutions of dynamical systems when a control parameter is varied. However, since deriving the algebraic system containing the Fourier coefficients can be a highly cumbersome procedure, the classical HBM is often limited to polynomial (quadratic and cubic) nonlinearities and/or a few harmonics. Several variations on the classical HBM, such as the incremental HBM or the alternating frequency/time domain HBM, have been presented in the literature to overcome this shortcoming. Here, we present an alternative approach that can be applied to a very large class of dynamical systems (autonomous or forced) with smooth equations. The main idea is to systematically recast the dynamical system in quadratic polynomial form before applying the HBM. Once the equations have been rendered quadratic, it becomes obvious to derive the algebraic system and solve it by the so-called ANM continuation technique. Several classical examples are presented to illustrate the use of this numerical approach

Since the founding theory established by G. Floquet more than a hundred years ago, computing the stability of periodic solutions has given rise to various numerical methods, mostly depending on the way the periodic solutions are themselves determined, either in the time domain or in the frequency domain. In this paper, we address the stability analysis of branches of periodic solutions that are computed by combining a pure Harmonic Balance Method (HBM) with an Asymptotic Numerical Method (ANM). HBM is a frequency domain method for determining periodic solutions under the form of Fourier series and ANM is continuation technique that relies on high order Taylor series expansion of the solutions branches with respect to a path parameter. It is well established now that this HBM-ANM combination is efficient and reliable, provided that the system of ODE is first of all recasted with quadratic nonlinearities, allowing an easy manipulation of both the Taylor and the Fourier series. In this context, Hill's method, a frequency domain version of Floquet theory, is revisited so as to become a by-product of the HBM applied to a quadratic system, allowing the stability analysis to be implemented in an elegant way and with good computing performances. The different types of stability changes of periodic solutions are all explored and illustrated through several academic examples, including systems that are autonomous or not, conservative or not, free or forced.