Anomalous Period-Doubling Bifurcation of the Elliptic Fixed Point in the Area-Preserving Maps

Abstract

In 1968, Moser reported a new bifurcation through which the periodic orbits with period-3 or -4 appear. At present, this bifurcation is called the anomalous rotation bifurcation (ARB). The examples of ARB have been already known. Why the anomalous period-doubling bifurcation (APDB) of the elliptic fixed point does not happen in the area-preserving maps? In order to answer this question, we introduce the area preserving map T defined by Cⁿ (n ≥ 1) mapping function and derive the conditions that APDB happens.