Dominance of Milnor attractors in globally coupled dynamical systems with more than 7+/-2 degrees of freedom.

The prevalence of Milnor attractors has recently been reported in a class of high-dimensional dynamical systems. We study how this prevalence depends on the number of degrees of freedom by using a globally coupled map and show that the basin fraction of Milnor attractors increases drastically around 5-10 degrees of freedom, saturating for higher numbers of degrees of freedom. It is argued that this dominance of Milnor attractors in the basin arises from a combinatorial explosion of the basin boundaries. In addition, the dominance is also found in a system without permutation symmetry, i.e., a coupled dynamical system of nonidentical elements. Possible relevance to the magic number 7+/-2 in psychology is briefly discussed.