Recent investigations on the bifurcations in switching circuits have shown
that many atypical bifurcations can occur in piecewise smooth maps which can
not be classified among the generic cases like saddle-node, pitchfork or Hopf
bifurcations occurring in smooth maps. In this paper we first present
experimental results to establish the need for the development of a theoretical
framework and classification of the bifurcations resulting from border
collision. We then present a systematic analysis of such bifurcations by
deriving a normal form --- the piecewise linear approximation in the
neighborhood of the border. We show that there can be eleven qualitatively
different types of border collision bifurcations depending on the parameters of
the normal form, and these are classified under six cases. We present a
partitioning of the parameter space of the normal form showing the regions
where different types of bifurcations occur. This theoretical framework will
help in explaining bifurcations in all systems which can be represented by two
dimensional piecewise smooth maps.Comment: Revised version of earlier submission. 11 pages. Gnuzipped postscript
including figures. To be published in Physical Review E, 1 April 199
