The Bundle Plot: Evolution of Symbolic Space under the System parameter Changes

This is a topological dynamics perspective on the full bifurcation unfolding in unimodal mappings. We present a bundle structure, visualized as a bundle plot, to show the evolution of symbolic space as we vary a system parameter. The bundle plot can be viewed as a limit process of an assignment plot, which are lines assignments between points from two dynamical systems. Such line assignments are determined by a commuter, which is a coordinates transformation function that satisfies a commuting relationship but not necessary a homeomorphism. The bundle structure is studied with understanding of implication from system’s qualitative changes. In addition, the case of the bundle plot with higher dimensional parameter variation is also considered. A main concern in the bundle plot is a special structure, called “joint”, which determines a critical value of the parameter where the kneading sequence becomes periodic.