Delay differential equations and dynamical systems
Stability and bifurcations of equilibria in a multiple-delayed differential equation
SIAM Journal on Applied Mathematics
A new method for computing delay margins for stability of linear delay systems
Systems & Control Letters
Rational Krylov: A Practical Algorithm for Large Sparse Nonsymmetric Matrix Pencils
SIAM Journal on Scientific Computing
The Quadratic Eigenvalue Problem
SIAM Review
SOAR: A Second-order Arnoldi Method for the Solution of the Quadratic Eigenvalue Problem
SIAM Journal on Matrix Analysis and Applications
Stability criteria for certain third-order delay differential equations
Journal of Computational and Applied Mathematics
Technical communique: On stability of second-order quasi-polynomials with a single delay
Automatica (Journal of IFAC)
Extended Kronecker Summation for Cluster Treatment of LTI Systems with Multiple Delays
SIAM Journal on Control and Optimization
Structured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations
SIAM Journal on Matrix Analysis and Applications
Vector Spaces of Linearizations for Matrix Polynomials
SIAM Journal on Matrix Analysis and Applications
Efficient computation of characteristic roots of delay differential equations using LMS methods
Journal of Computational and Applied Mathematics
Complete stability robustness of third-order LTI multiple time-delay systems
Automatica (Journal of IFAC)
EURASIP Journal on Bioinformatics and Systems Biology - Special issue on network structure and biological function: Reconstruction, modelling, and statistical approaches
Automatica (Journal of IFAC)
Node localization algorithm based on matrix pencil for wireless sensor network
NTMS'09 Proceedings of the 3rd international conference on New technologies, mobility and security
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In this work we present a new method to compute the delays of delay-differential equations (DDEs), such that the DDE has a purely imaginary eigenvalue. For delay-differential equations with multiple delays, the critical curves or critical surfaces in delay space (that is, the set of delays where the DDE has a purely imaginary eigenvalue) are parameterized. We show how the method is related to other works in the field by treating the case where the delays are integer multiples of some delay value, i.e., commensurate delays. The parameterization is done by solving a quadratic eigenvalue problem which is constructed from the vectorization of a matrix equation and hence typically of large size. For commensurate delay-differential equations, the corresponding equation is a polynomial eigenvalue problem. As a special case of the proposed method, we find a closed form for a parameterization of the critical surface for the scalar case. We provide several examples with visualizations where the computation is done with some exploitation of the structure of eigenvalue problems.