Fig.9: Initial conditions with certainty (left) and uncertainty (right).
The concept of fractal gives a viewpoint for objects in the nature based on the notion of self-similarity. In this section, we explain how to estimate dimension of a fractal basin boundary. Then the qualitative changes in basin structure illustrated in the previous section is quantitatively characterized with variation of fractal dimension. The method with uncertainty exponents was proposed by Ref.[12].
For a given initial condition, take a 2D open circle centered at the initial condition with radius d. We consider several points on the circle as neighbor points of the initial condition. If the destinations of the neighbor points match with the final state of the initial point as illustrated in Fig.9(left), then it is called a point with certainty. Otherwise, as illustrated in Fig. 9(right), it is called a point with uncertainty. For many initial conditions, we count the number of uncertainty points. Namely, it represents how many disks are needed to cover the basin boundary. If the rate of the uncertainty points scales with the radius d, then the boundary has a scaling property. The differential of the fitting line in the log-log plot provides the uncertainty exponent [12]. The capacity dimension of the basin boundary is given by 2 minus the uncertainty exponents.
Figure 10 indicates the scaling property of the basin boundary in two coupled chaotic maps. The linear fitting in a log-log plot yields the uncertainty exponent and the dimension of the basin boundary. A basin boundary embedding a chaotic repeller in a 2D map typically has dimension between 1 and 2.
