Continuation of Low-Dimensional Invariant Subspaces in Dynamical Systems of Large Dimension

Abstract

We present a continuation method for low-dimensional invariant subspaces of a parameterized family of large and sparse real matrices. Such matrices typically occur when linearizing about branches of steady states in dynamical systems that are obtained by spatial discretization of time-dependent PDE’s. The main interest is in subspaces that belong to spectral sets close the imaginary axis. Our continuation procedure provides bases of the invariant subspaces that depend smoothly on the parameter as long as the continued spectral subset does not collide with another eigenvalue. Generalizing results from [32] we show that this collision generically occurs when a real eigenvalue from the continued spectral set meets another eigenvalue from outside to form a complex conjugate pair. Such a situation relates to a turning point of the subspace problem and and we develop a method to inflate the subspace at such points.