Sensitivities and linear stability analysis around a double-zero eigenvalue

A general, multiparameter system admitting a double zero eigenvalue at a cr itical equilibrium point is considered. ri sensitivity analysis of the crit ical eigenvalues is performed to explore the neighborhood of the critical p oint in the parameter space. Because the coalescence of the eigenvalues imp lies that the Jacobian matrix is defective (or nilpotent), well-suited tech niques of perturbation analysis must be employed to evaluate the eigenvalue s and the eigenvector sensitivities. Different asymptotic methods are used, based on perturbations both of the eigenvalue problem and the characterist ic equation. The analysis reveals the existence of a generic (nonsingular) case and of a nongeneric (singular) case. However, even in the generic rase , a codimension-1 subspace exists in the parameter spare on which a singula rity occurs. By the use of the relevant asymptotic expansions, linear stabi lity diagrams are built up, and different bifurcation mechanisms (divergenc e-Hopf, double divergence, double divergence-Hopf, degenerate Hopf) are hig hlighted. The problem of finding 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 clarification of the geometrical meaning of the unfolding parameters, which has been discussed in literature for the Takens-Bogdanova bifurcation. Fin ally, a mechanical system loaded by nonconservative forces and exhibiting a double-zero bifurcation is studied as an example.