  The appearance of infinitely-many period-doubling cascades is one of the most
prominent features observed in the study of maps depending on a parameter. They
are associated with chaotic behavior, since bifurcation diagrams of a map with
a parameter often reveal a complicated intermingling of period-doubling
cascades and chaos. Period doubling can be studied at three levels of
complexity. The first is an individual period-doubling bifurcation. The second
is an infinite collection of period doublings that are connected together by
periodic orbits in a pattern called a cascade. It was first described by
Myrberg and later in more detail by Feigenbaum. The third involves infinitely
many cascades and a parameter value $\mu_2$ of the map at which there is chaos.
We show that often virtually all (i.e., all but finitely many) ``regular''
periodic orbits at $\mu_2$ are each connected to exactly one cascade by a path
of regular periodic orbits; and virtually all cascades are either paired --
connected to exactly one other cascade, or solitary -- connected to exactly one
regular periodic orbit at $\mu_2$. The solitary cascades are robust to large
perturbations. Hence the investigation of infinitely many cascades is
essentially reduced to studying the regular periodic orbits of $F(\mu_2,
\cdot)$. Examples discussed include the forced-damped pendulum and the
double-well Duffing equation.
