Detection of Hopf points by counting sectors in the complex plane

Abstract

Let \(A(\alpha)\) (\(\alpha\) real) be a family of real \(n\) by \(n\) matrices. A value \(\alpha^0\) of \(\alpha\) is called a Hopf value if \(A(\alpha^0)\) has a conjugate pair of purely imaginary eigenvalues \(\pm {\rm i}\omega^0\), \(\omega^0 > 0\). We describe a technique for detecting Hopf values based on the evolution of the Schur complement of \(A-zI\) in a bordered extension of \(A-zI\) where \(z\) varies along the positive imaginary axis of the complex plane. We compare the efficiency of this method with more obvious methods such as the use of the QR algorithm and of the determinant function of \(A\) as well as with recent work on the Cayley transform. In particular, we show the advantages of the Schur complement method in the case of large sparse matrices arising in dynamical problems by discretizing boundary value problems. The Hopf values of the Jacobian matrices are important in this setting because they are related to the Hopf bifurcation phenomenon where steady state solutions bifurcate into periodic solutions.