Bifurcation theory and harmonic content of oscillations

Abstract

The harmonic content of periodic solutions in ODEs is obtained using standard techniques of harmonic balance and the fast Fourier transform (FFT). For the first method, the harmonic content is attained in the vicinity of the Hopf bifurcation condition where a smooth branch of oscilations is born under the variation of a distinguished parameter. The second technique is applied directly to numerical simulation, which is assumed to be the correct solution. Although the first method is local, it provides an excellent tool to characterize the periodic behavior in the unfoldings of other more complex singularities, such as the double Hopf bifurcation (DHB). An example with a DBH is analyzed with this methodology and the FFT algorithm.