Period-doubling cascades galore

Period-doubling cascades for the quadratic family are familiar to every student of elementary dynamical systems. Cascades are also often observed in both numerical and scientific experiments. Yet in all but the simplest cases, very little is known about why they exist. When there is one cascade, there are almost always infinitely many. Further, there has been no systematic way of distinguishing the different types of cascades. In this paper, we explain why certain families of maps both in one dimension and higher dimensions have cascades and develop a method for counting them. The periods of the orbits in a cascade are {k, 2k, 4k, 8k . . . } for some k, and we refer to such a cascade as a period-k cascade. We refer to cascades which persist for all large parameters as unbounded cascades. We are able to enumerate the set of unbounded period-k cascades which occur for certain classes of families. The following example will give the flavor of our approach. Consider the map F (λ, x) = λ− x + g(λ, x), where λ and x ∈ R, and g : R × R → R, and consider its (periodic) orbits as λ is varied. For every generic smooth C1-bounded function g, the map F always has exactly the same number of period-k cascades as occur in the case g ≡ 0. That is, cascades are generically robust in the sense that for a residual set of g, there is no way to destroy them. We show that the results are not restricted to quadratic families, nor are they restricted to one dimension. We also give new topological results on the space of orbits in R × RN under the Hausdorff metric. We show that generically, after discarding “flip” orbits, each connected component is a one-manifold. Each unbounded component in this space of “nonflip” orbits must contain a cascade. 1 ar X iv :0 90 3. 36 13 v1 [ m at h. D S] 2 1 M ar 2 00 9