  We analyze a one-dimensional piecewise continuous discrete model proposed
originally in studies on population ecology. The map is composed of a linear
part and a power-law decreasing piece, and has three parameters. The system
presents both regular and chaotic behavior. We study numerically and, in part,
analytically different bifurcation structures. Particularly interesting is the
description of the abrupt transition order-to-chaos mediated by an attractor
made of an infinite number of limit cycles with only a finite number of
different periods. It is shown that the power-law piece in the map is at the
origin of this type of bifurcation. The system exhibits interior crises and
crisis-induced intermittency.
