Bifurcations and chaos in converters . Discontinuous vector fields and singular Poincaré maps

Basic geometrical and topological features are described for discontinuous systems modelling power converters. The global Poincaré map considered arises naturally from the sampling process in the oscillatory forced system. It is shown that this map belongs to the class of two-dimensional invertible continuous but only piecewise-smooth maps. It also contains a singular point at which critical curves accumulate. A method is found to compute analytically the characteristic multipliers of a periodic orbit, giving a powerful tool to obtain the values for smooth and non-smooth bifurcations. The images of the regions of multiple crossings are studied geometrically and then numerical computation allows one to deduce the existence of a Smale horseshoe mechanism in the map and also to obtain chaotic motion. Finally, the existence of a chaotic attractor is justified with the addition of 5T -recurrent behaviour near a singular point. (Some figures in this article appear in colour in the electronic version; see www.iop.org) AMS classification scheme numbers: 58F14, 58F13, 70K10, 70K50, 65Cxx