Matrices with rank deficiency two in eigenvalue problems and dynamical systems

Let $K(\alpha ,\beta )$ ($\alpha $, $\beta $ real) be a family of real square matrices. Several computational problems are equivalent to the calculation of a pair $(\alpha ^0 ,\beta ^0 )$ of parameter values for which $K(\alpha ^0 ,\beta ^0 )$ has rank deficiency 2. Among these problems are the computation of a conjugate pair of complex eigenvalues of a given real matrix, the computation of a Hopf bifurcation point on a branch of stationary solutions to a parametrized differential equation, and the computation of a Takens–Bogdanov point in the two-dimensional solution manifold of a set of nonlinear equations.Developing ideas of Griewank and Reddien, the authors define scalar functions $g_1 (\alpha ,\beta ),g_2 (\alpha ,\beta )$ that vanish in $(\alpha ^0 ,\beta ^0 )$. A nondegeneracy condition NDC, which expresses the fact that $(\alpha ^0 ,\beta ^0 )$ is in a natural sense an isolated point in $(\alpha ,\beta )$-space, is introduced. It is proved that under certain conditions on the family $K(\alpha ,\be...