Large sparse eigenvalue solvers for the determination of Hopf bifurcations of nonlinear dynamical systems

We review methods for computing the eigenvalues of a matrix pair near the imaginary axis. An application is the stability analysis of dynamical systems. During the last two decades, iterative solvers for computing a few eigenvalues of the nonsymmetric algebraic eigenvalue problem have been developed. Best known are Krylov methods, in particular the Arnoldi method. We review this method on the basis of the application of the determination of eigenvalues near the imaginary axis. We show that this method is reliable if it is used carefully. In addition, we will discuss a new idea based on parameterized eigenvalue problems. We will discuss the direct computation of the values of the parameter that give rise to a pair of purely imaginary eigenvalues, rather than monitoring eigenvalues as a function of the parameter.status: publishe