Non-resonant double Hopf bifurcations: The complex case

Abstract

The analysis of the unfolding of a non-resonant double Hopf singularity is considered using the frequency domain and the normal form methodologies. A higher-order harmonic balance and the evaluation of Floquet multipliers are used to obtain accurate approximations of the limit cycles and its local bifurcations. This type of hybrid methodology using harmonic balance and normal forms gives a complementary view in the vicinity of the singularity. More specifically, some of the bifurcation curves arising in the unfolding of the double Hopf bifurcation are computed with great accuracy using the harmonic balance method compared to the classical normal form. However, this last method is able to predict the appearance of complex quasiperiodic behavior such as three-dimensional (3D) torus. Numerical simulations corroborate this observation.