Center Bifurcation for a Two-Dimensional Piecewise Linear Map

Abstract

It is already well known that the main bifurcation scenario which can be realized considering a business cycle model in dynamic context, is related to a fixed point losing stability with a pair of complex-conjugate eigenvalues. In the case in which such a model is discrete and defined by some smooth nonlinear fianctions, the Neimark-Sacker bifiarcation theorem can be used, described in the previous chapter. While for piecewise linear, or piecewise smooth, functions which are also quite often used for business cycle modeling, the bifurcation theory is much less developed. The purpose of this chapter is to describe a so-called center bifurcation occurring in a family of two-dimensional piecewise linear maps whose dynamic properties are, to our knowledge, not well known. Namely, we shall see that in some similarity to the Neimark-Sacker bifurcation occurring for smooth maps, for piecewise linear maps the bifurcation of stability loss of a fixed point with a pair of complex-conjugate eigenvalues on the unit circle can also result in the appearance of a closed invariant attracting curve homeomorphic to a circle. However, differently from what occurs in the smooth case, the closed invariant curve is not a smooth, but a piecewise linear set, appearing not in a neighborhood of the fixed point, as it may be very far from it. In fact, we shall see that the position of the closed invariant curve depends on the distance of the fixed point from the boundary of the region in which the linear map is defined (i.e., from what we shall call critical line LC-i),