<b>Complex Systems</b><br><i>Connecting Period-Doubling Cascades to Chaos</i>

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.