Torus birth bifurcations in a DC/DC converter

Considering a pulsewidth modulated dc/dc converter as an example, this paper describes a border-collision bifurcation that can lead to the appearance of quasi-periodicity in piecewise-smooth dynamical systems. We demonstrate how a two-dimensional torus can arise from a periodic orbit through a bifurcation in which two complex-conjugate Poincare characteristic multipliers jump abruptly from the inside to the outside of the unit circle. The torus may be ergodic or resonant. However, in both cases the diameter of the torus develops approximately linearly with the distance to the bifurcation point as opposed to the characteristic parabolic form of the well-known Neimark-Sacker bifurcation. The paper also considers the birth of a torus via a subcritical Neimark-Sacker bifurcation in the piecewise-smooth system. Particular emphasis is given to the development of resonance zones via border-collision bifurcations.