Scaling pattern of period doubling in four dimensions.

Abstract

The period-{ital M} ({ital M}=1 and 2) scaling pattern of period doubling in a symmetric four-dimensional volume-preserving map is studied. The period-doubling sequence repeats itself asymptotically from one bifurcation to the next in a period-1 bifurcation route, and to every other one in a period-2 bifurcation route. The parameter-scaling factors {gamma}{sub 1} and {gamma}{sub 2} in a bifurcation route depend on the bifurcation path. They are some combination of {delta}{sub 1} and {delta}{sub 2} (divergence rates from the period-{ital M} map of the renormalization transformation) and {delta}{sub 1}{sup {prime}} and {delta}{sub 2}{sup {prime}} (convergence rates to the period-{ital M} map). Therefore each bifurcation route is characterized by these four scaling factors. The values of {delta}{sub 1}, {delta}{sub 2}, {delta}{sub 1}{sup {prime}}, and {delta}{sub 2}{sup {prime}} are obtained by a numerical calculation and a renormalization analysis. We find that there are three kinds of period-1 scaling patterns and one kind of period-2 scaling pattern. {delta}{sub 1} and {delta}{sub 1}{sup {prime}} in any bifurcation route are the same as those in area-preserving maps; however, {delta}{sub 2} and {delta}{sub 2}{sup {prime}} depend on the bifurcation route.