Period doubling cascades of attractors: A prerequisite for horseshoes
This paper investigates the dynamical behavior as the horseshoe is being formed in dependence on a parameter \(\ell\). At the initial parameter value \((\ell =0)\), the rectangle and its image are disjoint; at \(\ell =1\), there is a fully developed horseshoe map. Denoting the stable variable x and representing any periodic orbit by an appropriate single point from this orbit we obtain a complicated schematic diagram, which demonstrates the process of a creation of the horseshoe, in (x,\(\ell)\) space. Connected networks of orbits in this diagram are referred to as components. The general result proved in the paper may be stated loosely as follows. Choose any saddle in the horseshoe \((\ell =1)\). Let k denote its period. Examine its component: the network of orbits connected to it. Then, necessarily, the component will contain attractors of minimum period k,2k,4k,.... That is, there will be a cascade of period doubling attractors connected to the saddle. All these must occur before the horseshoe is fully developed at \(\ell =1\). Further it is proved that each saddle orbit has its own cascade and that there are approximately \(2^{k-1}/k\) saddle orbits of period k in the horseshoe.
scientific article; zbMATH DE number 3944902
scientific article