On the bifurcation structure of the mean-field fluctuation in the globally coupled tent map systems

A theoretical analysis on the behavior of the globally coupled tent map systems based on the kinetics of the mean-field value is presented. We investigated the nature of the mean-field fluctuation in large system size limit, and uncovered the existence of a complicated bifurcation structure that leads the appearance of collective motion with the increase of the coupling strength. The way to the appearance of the collective motion sensitively depends on the gradient a of individual tent-maps. For a belonging to a set which is dense in (root 2, 2), the collective motion appears when the coupling strength exceeds a certain threshold value, while for a belonging to another dense set in (root 2, 2), any non-zero coupling will induce the collective motion. The dependence of the amplitude of mean-field fluctuation on the coupling strength is also obtained as a function of the parameter a and the coupling strength. Copyright (C) 1998 Elsevier Science B.V.