Matrices with Rank Deficiency Two in Eigenvalue Problems and Dynamical Systems

Abstract

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 $K(\alpha ,\beta )$ the Jacobian of $g_1 $, $g_2 $ with respect to $\alpha $, $\beta $ is nonsingular if and only if NDC holds. A Newton method to compute $(\alpha ^0 ,\beta ^0 )$ is then described. The problems mentioned above and some related ones are analysed in detail. The derived algorithms for the dynamical systems problems have the following features: (1) they are simple and natural, being based on linear algebra concepts only; (2) they treat the two parameters in a symmetric way; (3) they do not lead to formally large systems; and (4) NDC is expressed in terms independent of the particular problem. Numerical results are given which illustrate the quadratic convergence of the Newton algorithm during the computation of a Hopf bifurcation point arising in a model of a tubular reactor.