Multiple time scales analysis for 1:2 and 1:3 resonant Hopf Bifurcations

The postcritical behavior of a general n-dimensional system around a resonant double Hopf bifurcation is analyzed. Both cases in which the critical eigenvalues are in ratios of 1:2 and 1:3 are investigated. The Multiple Scale Method is employed to derive the bifurcation equations systematically in terms of the derivatives of the original vector field evaluated at the critical state. Expansions of the n-dimensional vector of state variables and of a three-dimensional vector of control parameters are performed in terms of a unique perturbation parameter ε, of the order of the amplitude of motion. However, while resonant terms only appear at the ε3-order in the 1:3 case, they already arise at the ε2-order in the 1:2 case. Thus, by truncating the analysis at the ε3-order in both cases, first or second-order bifurcation equations are respectively drawn, the latter requiring resort to the reconstitution prin- ciple. A two-degrees-of-freedom system undergoing resonant double Hopf bifurcations is studied. The complete postcritical scenario is analyzed in terms of the three control parameters and the asymptotic results are compared with exact numerical integrations for both resonances. Branches of periodic as well as periodically modulated solutions are found and their stability analyzed.