Continuation of quasi-periodic solutions with two-frequency Harmonic Balance Method

Abstract

The continuation of quasi-periodic solutions for autonomous or forced nonlinear systems is presented in this paper. The association of the Asymptotic Numerical Method, a robust continuation method, and a two-frequency Harmonic Balance Method, is performed thanks to a quadratic formalism. There is no need for a priori knowledge of the solution: the two pulsations can be unknown and can vary along the solution branch, and the double Fourier series are computed without needing a harmonic selection. A norm criterion on Fourier coefficients can confirm a posteriori the accuracy of the solution branch. On a forced system, frequency-locking regions are approximated, without blocking the continuation process. The continuation of these periodic solutions can be done independently. On an autonomous system an example of solution is shown where the number of Fourier coefficients is increased to improve the accuracy of the solution.