A discontinuous, two-parameter family of one-dimensional maps is shown to posses many of the features of the quadratic (“logistic”) map; for instance, a unique, twice differentiable maximum. However, paths may be chosen in the two-dimensional parameter space which give rise to a wealth of possible routes to chaos. In particular, paths may be found for the Feigenbaum route to chaos — period-doubling cascade of bifurcations — with a resulting single parameter in the map converging geometrically with a constant ω<δ=4.6692016…. Also, the approach to chaos may be more abrupt; for instance, from a stable periodic orbit of any period n directly to chaotic motion.
