Corner collision implies border-collision bifurcation

This paper analyses a so-called corner-collision bifurcation in piecewise-smooth systems of ordinary differential equations (ODEs), for which a periodic solution grazes with a corner of the discontinuity set. It is shown under quite general circumstances that this leads to a normal form that is to lowest order a piecewise-linear map. This is the first generic derivation from ODE theory of the so-called C-bifurcation (or border collision) for piecewise-linear maps. The result contrasts with the equivalent results when a periodic orbit grazes with a smooth discontinuity set, which has recently been shown to lead to maps that have continuous first derivatives but not second. Moreover, it is shown how to calculate the piecewise-linear map for arbitrary dimensional systems, using only properties of the single periodic trajectory undergoing corner collision. The calculation is worked out for two examples, including a model for a commonly used power electronic converter where complex dynamics associated with corner collision was previously found numerically, but is explained analytically here for the first time. (C) 2001 Elsevier Science B.V. All rights reserved.