A new numerical approach for the efficient computation of Floquet multipliers within the harmonic balance technique

We present a novel numerical approach to the computation of the relevant Floquet eigenvalues and eigenvectors associated to large-signal periodic limit cycles within the framework of the harmonic balance technique. The currently available harmonic balance computation techniques for the Floquet quantities involve a large generalised eigenvalue problem, that provides a large set of eigenvalues containing the required Floquet exponents and a replica of them with imaginary part augmented by multiple integers of the fundamental angular frequency (we call this the Floquet eigenvalue split set). In our proposal, numerical efficiency is attained transforming the large generalised eigenvalue problem into a sequence of second order nonlinear systems whose size is of the same order as the standard harmonic balance problem. Such transformation avoids the determination of the Floquet eigenvalue split set and allows for the calculation of a predefined subset of the relevant Floquet quantities. For the discussed example, an advantage of 3 orders of magnitude in computation time is obtained. An important added value is that through this technique the automatic tracing of bifurcation curves is easily implemented through continuation methods.