Universality in multidimensional maps

Abstract

Bifurcations of cubic nonlinear symplectic mappings in two and four dimensions are discussed. Here series of bifurcations are studied by direct numerical calculation and by a renormalisation procedure. It is shown that for period-doubling bifurcations one finds the universal exponent of the quadratic area-preserving map. Other exponents exist for higher multiplicities. The renormalisation transformation has a fixed line in parameter space with an end point. The latter implies that series of period-doubling bifurcations may break off.