Pseudospectra Analysis, Nonlinear Eigenvalue Problems, and Studying Linear Systems with Delays

Time delays occur naturally in many physical and social systems. Computer simulations of such systems require that models of these systems be stable, and perhaps even passive, if several such systems are to be joined together in the simulation. We present a visual procedure for studying the stability and passivity of such systems. This procedure uses ideas from pseudospectra analysis. It is applicable to systems of linear, delay-differential-algebraic equations. There are no a priori restrictions on the types or sizes of the delays. No approximations to the original system are made. All approximations are confined to the grid used in the visualization procedure, and the procedure parallelizes readily. We apply this procedure to the study of the stability and passivity of proposed models for simulations of the behavior of currents and voltages in packaged VLSI interconnects (wires and planes) in computers. Simulations are required to verify that internal electromagnetic fields do not significantly delay or distort circuit signals.