Fig.15: Scaling property of a basin boundary in the chaotic neural network.
The idea considered in the previous section is applied to an analysis of chaotic neural network [1]. When the chaotic neural network is used as an associative memory model, multiple memories are embedded in chaotic attractors. It exhibits a sudden onset of intermittent behaviors called itinerant memory dynamics similar to that in two coupled logistic maps. The mechanism of the global bifurcation has been clarified from the viewpoint of intersections of unstable manifolds [18, 9].
Before the bifurcation, there coexist four chaotic attractors symbolizing memories in 3D phase space. A typical orbit asymptotes to one of them depending on an initial condition. This corresponds to a recall of a memory. After the bifurcation, a typical orbit exhibits itinerant transitions between four regions as demonstrated in Fig.14. The behavior can be regarded as a history-dependent memory association.
In 3D phase space, it is difficult to observe basin structure, even if it is projected to 2D space. Thus, we assume that a basin boundary is fractal and estimate the uncertainty exponent. In fact, the linear fitting as shown in Fig. 15 indicates scaling property of the basin boundary. In a similar way, we obtain variation of the fractal dimension for change of the parameter kr by estimating the uncertainty exponent as shown in Fig. 16. The failure of the computation of the dimension at near kr=0.877 suggests the disappearance of the fractal basin boundary and the onset of the crisis-induced intermittency.
