Role of the Virtual Fixed Point in the Center Bifurcations in a Family of Piecewise Linear Maps

Abstract

In this work, we consider a continuous two-dimensional piecewise linear family of maps in which the fixed point and other cycles undergo a center bifurcation, the analogue of a Neimark–Sacker bifurcation, which has a particular structure when occurring in piecewise linear maps. Our goal is to determine and characterize the occurrence of a center bifurcation of supercritical or subcritical type. We show that for the fixed point, a determinant role is played by the virtual fixed point of the system. Also for other [Formula: see text]-cycles the virtual fixed point plays a determinant role, as well as the saddle cycles related to the cycle which undergoes a center bifurcation. A particular bifurcation point separating the two transitions, supercritical and subcritical, is related to a conservative system (the analogue of the Chenciner point in smooth maps).