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

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. We show that the predictor and the corrector step during continuation lead to bordered matrix equations of Sylvester type. For these equations we develop a bordered version of the Bartels-Stewart algorithm which allows to reduce the linear algebra to solving a sequence of bordered linear systems. The numerical techniques are illustrated by studies of the stability problem for traveling waves in parabolic systems, in particular for the Ginzburg-Landau and the FitzHugh-Nagumo equation.