Search for Periodic Orbits Appearing through Anomalous Rotation Bifurcation

In a lecture that J.K.Moser gave in 1968, he pointed out the possibility that the periodic orbit with period q=3 (period-3 orbit) appears through anomalous rotation bifurcation (ARB). There are two different types of rotation bifurcation, the ordinary rotation bifurcation (ORB) and ARB. In order to study ORB and ARB, the map Tm (yn+1=yn+a(xn−xnm), xn+1=xn+yn+1 (a≥0, m≥2)) is introduced. It is proved that the period-3 orbit in Tm (m≥2) and the period-4 orbit in Tm (m≥3) appear through ARB and the period-4 orbit in T2 appears through ORB.