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

Yoshihiro Yamaguchi¹ and Kiyotaka Tanikawa²

¹2-4-14, Kokubunjidai-chuo, Ichihara, Chiba 290-0193, Japan
²National Astronomical Observatory, Mitaka 181-8588, Japan
E-mail address: chaosfractal@icloud.com; tanikawa.ky@nao.ac.jp

(Received August 31, 2021; Accepted October 27, 2021)

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.

Keywords: Anomalous Rotation/Period-Doubling Bifurcation, Area-Preserving Map