On the existence of low-period orbits in n-dimensional piecewise linear discontinuous maps

Discontinuous maps occur in many practical systems, and yet bifurcation phenomena in such maps is quite poorly understood. In this paper, we report some important results that help in analyzing the border collision bifurcations that occur in n-dimensional discontinuous maps. For this purpose, we use the piecewise linear approximation in the neighborhood of the plane of discontinuity. Earlier, Feigin had made a similar analysis for general n-dimensional piecewise smooth continuous maps. In this paper, we extend that line of work for maps with discontinuity to obtain the general conditions of existence of period-1 and period-2 fixed points before and after a border collision bifurcation. The application of the method is then illustrated using a specific example of a two-dimensional discontinuous map.