Sensitivities and linear stability analysis around a double zero eigenvalue

A general, multiparameter system admitting a double-zero eigenvalue at a critical equilibrium point is consid- ered. A sensitivity analysis of the critical eigenvalues is performed to explore the neighborhood of the critical point in the parameter space. Because the coalescence of the eigenvalues implies that the Jacobian matrix is defective (or nilpotent), well-suited techniques of perturbation analysis must be employed to evaluate the eigenvalues and the eigenvector sensitivities. Different asymptotic methods are used, based on perturbations both of the eigenvalue problem and the characteristic equation. The analysis reveals the existence of a generic (nonsingular) case and of a nongeneric (singular) case. However, even in the generic case, a codimension-1 subspace exists in the parameter space on which a singularity occurs. By the use of the relevant asymptotic expansions, linear stability diagrams are built up, and different bifurcation mechanisms (divergence– Hopf, double divergence, double divergence– Hopf, degenerate Hopf) are highlighted. The problem of nding a unique expression uniformly valid in the whole space is then addressed. It is found that a second-degree algebraic equation governs the behavior of the critical eigenvalues. It also permits clari cation of the geometrical meaning of the unfolding parameters, which has been discussed in literature for the Takens– Bogdanova bifurcation. Finally, a mechanical system loaded by nonconservative forces and exhibiting a double-zero bifurcation is studied as an example.