New universal bifurcation scenario in one-dimensional trimodal maps

By means of star products and high precision numerical calculation, an abnormal phenomenon is found in period-p-tupling bifurcation processes in one-dimensional trimodal maps. A route of transition to chaos, presented by a right-associative non-normal star product, breaks the Feigenbaum's metric universality, namely, the conventional Feigenbaum's successive rates exhibit a strong divergence. To overcome the divergence, an approximate scheme of accelerating convergence is proposed; and the Feigenbaum scenario is included as a special case in the new bifurcation scenario. It will provide access to understanding non-normal star products and their corresponding renormalization.