Sensitivities and Linear Stability Analysis Around a Double-Zero Eigenvalue

Abstract

A general, multiparameter system admitting a double-zero eigenvalue at a critical equilibrium point is considered. 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 arebuiltup, and different bifurcation mechanisms (divergence‐Hopf, doubledivergence, doubledivergence ‐Hopf, degenerateHopf )arehighlighted. The problem of e nding a uniqueexpression uniformly valid in the wholespaceis then addressed. It isfound that a second-degreealgebraic equation governsthebehaviorofthecriticaleigenvalues. It also permits clarie 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.