  The presence of a period-doubling cascade in dynamical systems that depend on
a parameter is one of the basic routes to chaos. It is rarely mentioned that
there are virtually always infinitely many cascades whenever there is one. We
report that for one- and two-dimensional phase space, in the transition from no
chaos to chaos -- as a parameter is varied -- there must be infinitely many
cascades under some mild hypotheses. Our meaning of chaos includes the case of
chaotic sets which are not attractors. Numerical studies indicate that this
result applies to the forced-damped pendulum and the forced Duffing equations,
viewing the solutions once each period of the forcing. We further show that in
many cases cascades appear in pairs connected (in joint parameter-state space)
by an unstable periodic orbit. Paired cascades can be destroyed or created by
perturbations, whereas unpaired cascades are conserved under even significant
perturbations.
