Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations

Several recent methods used to analyze asymptotic stability of delay-differential equations (DDEs) involve determining the eigenvalues of a matrix, a matrix pencil or a matrix polynomial constructed by Kronecker products. Despite some similarities between the different types of these so-called matrix pencil methods, the general ideas used as well as the proofs differ considerably. Moreover, the available theory hardly reveals the relations between the different methods. In this work, a different derivation of various matrix pencil methods is presented using a unifying framework of a new type of eigenvalue problem: the polynomial two-parameter eigenvalue problem, of which the quadratic two-parameter eigenvalue problem is a special case. This framework makes it possible to establish relations between various seemingly different methods and provides further insight in the theory of matrix pencil methods. We also recognize a few new matrix pencil variants to determine DDE stability. Finally, the recognition of the new types of eigenvalue problem opens a door to efficient computation of DDE stability

A structure preserving shift-invert infinite Arnoldi algorithm for a class of delay eigenvalue problems with Hamiltonian symmetry

In this work we consider a class of delay eigenvalue problems that admit a spectrum similar to that of a Hamiltonian matrix, in the sense that the spectrum is symmetric with respect to both the real and imaginary axis. More precisely, we present a method to iteratively approximate the eigenvalues of such delay eigenvalue problems closest to a given purely real or imaginary shift, while preserving the symmetries of the spectrum. To this end the presented method exploits the equivalence between the considered delay eigenvalue problem and the eigenvalue problem associated with a linear but infinite-dimensional operator. To compute the eigenvalues closest to the given shift, we apply a specifically chosen shift-invert transformation to this linear operator and compute the eigenvalues with the largest modulus of the new shifted and inverted operator using an (infinite) Arnoldi procedure. The advantage of the chosen shift-invert transformation is that the spectrum of the transformed operator has a "real skew-Hamiltonian"-like structure. Furthermore, it is proven that the Krylov space constructed by applying this operator, satisfies an orthogonality property in terms of a specifically chosen bilinear form. By taking this property into account during the orthogonalization process, it is ensured that even in the presence of rounding errors, the obtained approximation for, e.g., a simple, purely imaginary eigenvalue is simple and purely imaginary. The presented work can thus be seen as an extension of [V. Mehrmann and D. Watkins, "Structure-Preserving Methods for Computing Eigenpairs of Large Sparse Skew-Hamiltonian/Hamiltonian Pencils", SIAM J. Sci. Comput. (22.6), 2001], to the considered class of delay eigenvalue problems. Although the presented method is initially defined on function spaces, it can be implemented using finite dimensional linear algebra operations

Neutral Equations of Mixed Type

In this dissertation we consider neutral equations of mixed type. In particular, we con- sider the associated linear Fredholm theory and nerve fiber models that are written as systems of neutral equations of mixed type. In Chapter 2, we extend the existing Fredholm theory for mixed type functional differential equations developed by Mallet-Paret to the case of implicitly defined mixed type functional differential equations. In Chapter 3, we apply the theory to an example arising from modeling signal prop- agation in nerve fibers. In this two-dimensional system, we rely on the Lyapunov- Schmidt method to demonstrate the existence of traveling wave solutions. With the aid of numerical computations, a saddle-node bifurcation was detected. In Chapter 4, we consider an extension of the parallel nerve fiber model examining in Chapter 3 and present the results of a numerical study. In this chapter, an additional form of coupling is examined not considered in the model from Chapter 3. This second type of coupling may be excitatory or inhibitory depending on the sign of the coupling parameter. Within a continuation framework, we employ a pseudo-spectral approach utilizing Chebyshev polynomials as basis functions. The chebfun package, consisting of Chebyshev tools, was utilized to manipulate the polynomials

Localization theorems for nonlinear eigenvalue problems

Let T : \Omega \rightarrow \bbC^{n \times n} be a matrix-valued function that is analytic on some simply-connected domain \Omega \subset \bbC. A point $\lambda \in \Omega$ is an eigenvalue if the matrix $T(\lambda)$ is singular. In this paper, we describe new localization results for nonlinear eigenvalue problems that generalize Gershgorin's theorem, pseudospectral inclusion theorems, and the Bauer-Fike theorem. We use our results to analyze three nonlinear eigenvalue problems: an example from delay differential equations, a problem due to Hadeler, and a quantum resonance computation.Comment: Submitted to SIMAX. 22 pages, 11 figure