The bandcount increment scenario. III. Deformed structures

Abstract

Bifurcation structures in two-dimensional parameter spaces formed by chaotic attractors alone are still a long way from being understood completely. In a series of three papers, we investigated the chaotic domain without periodic inclusions for a map, which is considered by many authors as some kind of one-dimensional canonical form for discontinuous maps. In Part I, the basic structures in the chaotic region are explained by the bandcount increment scenario. In Part II, fine self-similar substructures nested into the bandcount increment scenario are explained by the bandcount-adding and -doubling scenarios, nested into each other ad infinitum. Hereby, we fixed in both previous parts one of the parameters to a non-generic value, and studied the remaining two-dimensional parameter subspace. In this Part III, finally we investigated the structures under variation of this third parameter. Remarkably, this step is the most important with respect to practical applications, since it cannot be expected that these operate exactly at the previously investigated specific value.