Speaker: Evelyn Sander (Mathematical Sciences, George Mason University)
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. This talk describes recent research in which we are able to link cascades and chaos in a new way.
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. The third level of complexity involves a transition from simple behavior at one parameter value through infinitely-many cascades until it reaches a larger parameter value at which there is chaos. In this talk, we describe recent work using topological methods showing that often virtually all (i.e., all but finitely many) ``regular'' periodic orbits in the chaotic regime 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 in the chaotic regime Furthermore, solitary cascades are robust to large perturbations. Hence the investigation of infinitely many cascades is essentially reduced to studying the regular periodic orbits of a one-parameter family. Examples discussed include the forced-damped pendulum and the double-well Duffing equation. This talk will not assume prior knowledge of either topology or cascades.