Fig.12: Variation of fractal dimension of the basin boundary.
The qualitative changes in the basin structure as a scenario for the crisis can be characterized by variation of the fractal dimension. Figure 12 shows variation of dimension of the fractal basin boundary as the parameter a is varied. At the genesis of the fractal basin boundary, the dimension is approximately two. The jump of the dimension at near a=1.63 is due to the emergence of an attractor on the diagonal. The decay of the dimension terminates at the bottom, which may correspond to the change from simply connected immediate basin to multiply one. At the final crisis, the computation of the dimension fails due to a disappearance of the fractal basin boundary.
The development of attractors and basin bifurcations with change of a can be confirmed in the animation in Fig. 13.
