This paper reports on the latest developments in taxonomy theory originally developed for dynamic systems represented by ordinary differential equations constrained by algebraic equations (DAEs). It gives some new results on the behavior of a second order state constrained system. It shows that the interplay between an unstable system and the hard limit can result generically in stable non-smooth limit cycles. These results are of particular importance for Hopf bifurcations which occur near the state limit since they point towards the possibility of stable operation even after a subcritical Hopf bifurcation has occured.

This paper presents a time-delay formulation of the large power system dynamics which includes modeling the fast distributed parameter transmission system dynamics from within the familiar phasor domain of the equipment dynamics. The notion of fast time-varying phasors proposed recently for the balanced power system leads to explicit solutions in the form of time-delayed algebraic equations for the transmission system dynamics which is represented by a large interconnected set of partial-differential wave equations, under certain assumptions.>

Connections between the theorems of Bussgang, Price, McMahon, Novikov, and Wiener are established, and new theorems are introduced for nonlinear functionals of Gaussian random processes by the use of functional calculus. In particular, Novikov's formula is extended to obtain an expression for the Wiener-Hermite kernels of a functional in terms of its functional derivatives.