Connecting period-doubling cascades to chaos

The appearance of infinitely-many period-doubling cascades is one of the most prominent features observed in the study of maps de-pending 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 µ2 of the map at which there is chaos. We show that often virtually all (i.e., all but finitely many) “regular ” periodic orbits at µ2 are each connected to exactly one cascade by a path of regular periodic orbits; and virtually all cas-cades are either paired – connected to exactly one other cascade, or solitary – connected to exactly one regular periodic orbit at µ2. The solitary cascades are robust to large perturbations. Hence the inves-tigation of infinitely many cascades is essentially reduced to studying the regular periodic orbits of F (µ2, ·). Examples discussed include the forced-damped pendulum and the double-well Duffing equation.