Effects of the Gear Discretization Order on the Computation of Bifurcation Boundaries

Abstract

We extend a discrete-time approach for the analysis of the steady-state and local stability of nonlinear circuits, based on Gear discretizations, to compute the bifurcation boundaries of periodically forced nonlinear circuits. A bifurcation point may be detected by following a limit cycle solution as a function of a parameter until an eigenvalue crosses the unit circle. However, efficiency is improved by adding an extra equation that places this eigenvalue on the unit circle. This allows tracing directly the boundaries of distinct operation regions in a parameter space. As an application example, we study the fold, flip and Neimark-Sacker bifurcation boundaries of a forced van der Pol oscillator. While the effects of the discretization order on the computation of period-1 solutions are well-known, we investigate its effects on the accuracy of the bifurcation points and the associated period-p solutions. We conclude that there may be significant errors in computing bifurcation points and period-p solutions when the usual first and second order Gear discretizations are used.