Identification of symmetry breaking and a bifurcation sequence to chaos in single particle dynamics in magnetic reversals

Regular and stochastic behaviour in single particle orbits in static magnetic reversals have wide application in laboratory and astrophysical plasmas and have been studied extensively. In a simple magnetic reversal of the form B=B0(f(z),0,b1) with an odd function f(z) providing the reversing field component and a constant b1 providing linking field component, the system has three degrees of freedom but only two global (exact) constants of the motion, namely the energy, h, and the canonical momentum in the y-axis, Py. Hence, the system is non-integrable and the particle motion can, under certain conditions, exhibit chaotic behaviour. Here we consider the dynamics when a constant shear field, b2, is added so that B=B0(f(z),b2,b1). In this case, the form of the potential changes from quadratic to velocity dependent. We use numerically integrated trajectories to show that the effect of the shear field is to break the symmetry of the system so that the topology of the invariant tori of regular orbits is changed. This has several important consequences: (1) the change in topology cannot be transformed away in the case of b2≠0 and hence the system cannot be transformed back to the more easily understood shear free case (b2=0); (2) invariant tori take the form of nested Moebius strips in the presence of the shear field. The route to chaos is via bifurcation (period doubling) of the Moebius strip tori.