Erik Mosekilde

Quasi-periodicity and border-collision bifurcations in a DC-DC converter with pulsewidth modulation

The paper considers the dynamics of a dc-dc converter with pulsewidth modulation. The typical scenario for the transition to chaos in such systems proceeds via quasi-periodicity, resonance cycles, and torus destruction. Detailed bifurcation analysis shows that the resonance solutions arise via border-collision bifurcations (BCB) on a two-dimensional torus. The arrangement of the resonance domains within the parameter plane is related to the Farey series, and their internal structure is described. It is shown that transitions to chaos mainly occur through finite sequences of BCB. Some other possible causes of complex dynamics are considered, including a subcritical Neimark-Sacker (Andronov-Hopf) bifurcation and the associated hysteretic phenomena.

Torus-Bifurcation Mechanisms in a DC/DC Converter With Pulsewidth-Modulated Control

Pulse-modulated converter systems play an important role in modern power electronics. However, by virtue of the complex interplay between ordinary (smooth) and so-called border-collision bifurcations generated by the switching dynamics, the changes in behavior that can occur in multilevel converter systems under varying operational conditions still remain to be explored in full. Considering the dynamics of a three-level dc/dc-converter, we demonstrate a number of new scenarios for the birth or destruction of resonant and ergodic tori. One scenario involves the formation of a doubled-layered torus structure around a stable focus point through three subsequent border-collision fold bifurcations. Another scenario replaces one of the fold bifurcations by a global bifurcation. In both of these scenarios, the basic mode of the converter remains stable while other modes grow up and bifurcate around it. We also illustrate the subcritical birth of both an ergodic and a resonance torus from the basic operational mode.

Multistability and Torus Reconstruction in a DC–DC Converter With Multilevel Control

By virtue of their limited size and relatively low costs, multilevel dc-dc converters have come to play an important role in modern industrial power supply systems. When operating in a regime of high corrector gain, such converters can display a variety of new dynamic phenomena associated with the appearance of additional oscillatory modes. Starting in a state where four pairs of stable and unstable period-6 cycles coexist with the basic period-1 cycle, the paper describes a sequence of smooth and nonsmooth bifurcations through which the cycles and their basins of attraction transform as the output voltage is increased. The paper also describes the birth of a multilayered resonance torus through a transverse pitchfork bifurcation of the saddle cycle on an ordinary resonance torus.