Impasse points. Part I: Numerical aspects

Impasse point is an important phenomenon found in many non-linear circuits and systems. Among other things, the presence of an impasse point Q implies that the circuit model is defective and must be remodelled by augmenting it with parasitic inductances and/or capacitances at appropriate locations in order to predict the bifurcation from slow to rapid motions (jump phenomenon) widely observed in practice. the presence of an impasse point Q also implies that a numerical simulation of the associated system of implicit differential-algebraic equations would give rise to an extraneous and random small-amplitude oscillation in the vicinity of Q. the wave-form associated with this ‘fake’ oscillatory phenomenon depends on the error-controlled mechanism of the integration routine and can be detected using the results from this paper. Impasse points in high-order circuits (n > 2) need not be isolated. In fact, they often form an impasse set made of impasse curves, impasse surfaces, impasse manifolds, etc. In order to detect the impasse set associated with the implicit differential-algebraic system. An analytical characterization of an impasse point (X0,y0) is derived and proved to be equivalent to that of solving for the limit-points of an associated static bifurcation problem. Interpreting this as the parametric equation of a space curve in the y-space, an impasse point (X0,y0) is simply a ‘turning point’ of this curve.