Low-dimensional piecewise smooth maps with an unpredictable number of switching manifolds

It is well-known that piecewise-smooth systems are extremely challenging from a mathematical point of view, in particular because of the bifurcation phenomena caused by interactions between invariant sets and switching manifolds. As a first necessary step, the theory of piecewise-smooth systems has been developed for models with one switching manifold. However, this is not always sufficient for practical applications. In particular, a worldwide growing interest in renewable energy sources (such as solar photovoltaic and wind energy systems) as well as electric car drives led in the last years to an increasing interest in dynamics of power electronic DC/AC converters. However, modeling of DC/AC converters leads to piecewise smooth stroboscopic maps whose properties still remain to be examined in detail. A distinguishing feature of these maps is their extremely high (and practically unpredictable) number of switching manifolds. This causes several unusual bifurcations phenomena to occur. We report how the transition from the domain where the stroboscopic map has stable and globally attracting fixed points to the region of chaotic dynamics occurs via irregular cascades of border collisions. We show how some of these border collisions form complex patterns in the stable domain, and how smooth (pitchfork and flip) bifurcations of different fixed points form (under certain consitions) macroscopic patterns stratching across the overall bifurcation structure.