CENTER BIFURCATION FOR TWO-DIMENSIONAL BORDER-COLLISION NORMAL FORM

Abstract

In this work we study some properties associated with the border-collision bifurcations in a two-dimensional piecewise-linear map in canonical form, related to the case where a fixed point of one of the linear maps has complex eigenvalues and undergoes a center bifurcation when its eigenvalues pass through the unit circle. This problem is faced in several applied piecewise-smooth models, such as switching electrical circuits, impacting mechanical systems, business cycle models in economics, etc. We prove the existence of an invariant region in the phase space for parameter values related to the center bifurcation and explain the origin of a closed invariant attracting curve after the bifurcation. This problem is related also to particular border-collision bifurcations leading to such curves which may coexist with other attractors. We show how periodicity regions in the parameter space differ from Arnold tongues occurring in smooth models in case of the Neimark–Sacker bifurcation, how so-called dangerous border-collision bifurcations may occur, as well as multistability. We give also an example of a subcritical center bifurcation which may be considered as a piecewise-linear analogue of the subcritical Neimark–Sacker bifurcation.