Detailed Solution Curves and Bifurcation Boundaries of the Forced van der Pol Oscillator

Abstract

We make use of a discrete-time approach for the analysis of the steady-state and local stability of nonlinear circuits 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 directly places this eigenvalue on the unit circle. This permits us to directly trace the boundaries of distinct operating regions in a two parameter space. We study the fold, flip and Neimark-Sacker bifurcation boundaries of a forced van der Pol oscillator. We focus our attention on the dynamics, to our knowledge not previously reported, that appear near the zone where the period-2 operating region intersects the quasi-periodic solution boundary.