FRACTAL TRANSITION: HIERARCHICAL STRUCTURE AND NOISE EFFECT

Abstract

A Sierpinski gasket with continuous trajectories is presented as an example of the fractal transition that characterizes the behavior of dissipative dynamical systems excited by external temporal inputs. Using this example, we investigate the fractal transition from two points of views, i.e. a hierarchical structure and a noise effect. Depending on internal and external parameters, the structure can be geometrically classified as one of three types, i.e. totally disconnected, just-touching, and overlapping. For the totally disconnected structure, continuous trajectories and their starting points can be characterized by a definite hierarchical tree structure. Even for the just-touching and overlapping structure, a similar hierarchy exists. White noise contaminating the external inputs breaks the hierarchy. In particular, small clustered structures are sensitive to the noise. In such a case, the difference between trajectories and starting points is remarkable in the hierarchy.