N Group Analysis of Bifurcations . in Area

We study the universal .properties of period-doubling bifurcation sequences in twodimensional area-preserving maps. This paper begins with a description of the period-doubling process based on the work of Greene, MacKay, Vivaidi, and Feigenbaum [I] and others [2, 3, 4]. The scaling laws which the period-doubling process is found to obey are explained through a renormalization argument previously presented by Greene et al. [I], by Helleinan [5, 6] and others [7, S]. Tt.is argument assumes the existence of a universal mapping which reproduces itself under a combined line~x change of coordinates and composition with itself. We determine this function numerically, utilizing the principle of least action to perform the compositions. The effects of perturbations on this universal mapping are analyzed.