A harmonic-based method for computing the stability of periodic solutions of dynamical systems

In this Note, we present a harmonic-based numerical method to determine the local stability of periodic solutions of dynamical systems. Based on the Floquet theory and the Fourier series expansion (Hill method), we propose a simple strategy to sort the relevant physical eigenvalues among the expanded numerical spectrum of the linear periodic system governing the perturbed solution. By mixing the harmonic-balance method and asymptotic numerical method continuation technique with the developed Hill method, we obtain a purely-frequency based continuation tool able to compute the stability of the continued periodic solutions in a reduced computation time. To validate the general methodology, we investigate the dynamical behavior of the forced Duffing oscillator with the developed continuation technique.