Metric universalities and systems of renormalization group equations for bimodal maps

For bimodal maps the Feigenbaum’s renormalization group (RG) equation is generalized to a system of two independent RG equations, each corresponding to a critical point. A series of numerical solutions to systems of the RG equations are obtained, which explain the metric universalities of period-tripling and period-quadrupling bifurcations. For a special type of period-p-tupling bifurcations, it is found that the “collision” of the two critical points of bimodal maps leads to a quadratic relation between the two universal scaling factors.