Some properties of a two-dimensional piecewise-linear noninvertible map

Properties of a piecewise-linear noninvertible map of the plane are studied by using the method of critical curves (two-dimensional extension of the notion of critical point in the one-dimensional case). This map is of (Z(0) - Z(2)) type, i.e. the plane consists of a region without preimage, and a region giving rise to two rank one preimages. For the considered parameter values, the map has two saddle fixed points. The characteristic features of the ''mixed chaotic area'' generated by this map, and its bifurcations (some of them being of homoclinic and heteroclinic type) are examined. Such an area is bounded by the union of critical curves segments and segments of the unstable set of saddle cycles.