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 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 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 cascades 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 investigation 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.