Periodic orbits in a near Yang-Mills potential

Abstract

Abstract We study numerically the orbits in the Yang-Mills (YM) potential V = 1 2 x 2 y 2 and in the potentials of the general form V = 1 2 [ α ( x 2 + y 2 ) + x 2 y 2 ] . We found that the stable period-9 periodic orbit of the YM potential belongs to a family of orbits that bifurcates from a basic period-9 family of the general form of the potential V, when α is slightly above zero. This basic period-9 family and its bifurcations exist only up to a maximum value of α = α max . We calculate the Hénon stability index of these orbits. The pattern of the stability diagram is the same for all the symmetric orbits of odd periods 3,5,7,9 and 11, that we have found. We also found the stability diagrams for asymmetric orbits of period 2,3,4,5 which have again the same pattern. All these orbits are unstable for α=0 (YM potential) except for the stable orbits of period-9 and some orbits with multiples of 9 periods.