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. * 313 Fractals 1999.07:313-326. Downloaded from www.worldscientific.com by FLINDERS UNIVERSITY LIBRARY on 02/03/15. For personal use only.

Inverse Proportional Relationship Between Switching-Time Length and Fractal-Like Structure for Continuous Tracking Movement

“…In addition, as the results of the inverse proportional relationship D ∝ 1/T in this study are almost in agreement with the theoretical analysis reported in [Gohara & Okuyama, 1999b;, it is considered that the correlation dimension D 2 is inversely proportional to the switching-time length T . However, the following findings emerged when we analyzed the relationship.…”

“…Moreover, we confirmed the inverse proportional relationship between T and the fractal-like feature D 2 in human pointing movement [Hirakawa et al, 2016]. Our purpose was to develop these studies on continuous tracking movements and to examine the regular relationship between T and D 2 , which is estimated by the transition state and which occurs when a human switches several attractors stochastically, based on a theoretical model [Gohara & Okuyama, 1999b;. It is assumed that an attractor underlies a movement pattern in a nonautonomous dynamical system with an external input [Gohara & Okuyama, 1999a].…”

“…The fitting curve was D 2 = 0.178/T + 0.738 when the value of the residual sum of squares was the smallest. From the above theoretical analysis, we find that in the case of tracking movements performed by a human that require abrupt switching of several movement patterns, the correlation dimension D 2 is inversely proportional to only time T , as reported in the theoretical analysis by Gohara and Okuyama [1999b], , if we assume that λ is constant although we are not able to estimate this value of λ in this study. These results suggest that the value of the correlation dimension increases gradually as T decreases and that a regular relationship such as an inverse proportion D 2 ∝ 1/T exists between the switching-time length T and the corre lation dimension D 2 .…”

“…Hence, we perform the experiment in this study with the dissipative nonautonomous dynamical system theory [Gohara & Okuyama, 1999b;. We briefly survey this theory in this section.…”

“…Moreover, multiple comparisons revealed that the value of the correlation dimension for T = 2.00 s was lower than the value at other T s, the value for T = 1.00 s was lower this numerically expresses the state that humans become less able to track external inputs gradually as T becomes even shorter. Theoretically, the relationship between the switching-time length T and the correlation dimension D 2 suggested by Gohara and Okuyama [1999b], is…”