A paradigmatic system to study the transition from zero/Hopf to double-zero/Hopf bifurcation

A two-d.o.f. system experiencing codimension-three double-zero/Hopf bifurcation is considered. This is a special bifurcation which simultaneously involves a defective and a nondefective pair of critical eigenvalues, therefore, requiring a perturbation method specifically tailored on it. A nonstandard version of the multiple scale method is implemented, in which fractional power expansions, both for state-variables and time are used, and high-order arbitrary amplitudes are introduced. Bifurcation equations are obtained, governing the slow flow on the center manifold, which turns out to be tangent to the space spanned by the four critical eigenvectors. These are used to analyze the transition from codimension-three to codimension-two single-zero/Hopf bifurcations, occurring when the modulus of the damping is increased from small to order-one values. Bifurcation charts are obtained, displaying the role of quasi-periodic motions in the transition.