Nearly straight line trajectories of the standard map with twist

Abstract

Some trajectories of the standard group with twist appear to be 'nearly straight lines'. In order to understand the cause of this appearance, a two-dimensional Hamiltonian function with rotational symmetry derived from the map has been studied. For three-, four- or six-fold symmetry, the plane is tiled periodically by sets of parallel straight line energy contours joining the hyperbolic fixed points. For other rotational symmetries, periodic tiling of the plane is not possible, but in some cases there is a strong appearance of quasi-periodic tiling by energy contours corresponding to nearly straight line trajectories of the map. Such energy contours are associated with straight lines along which the variance of the Hamiltonian is a local minimum. For five- and eight-fold symmetry the local minimum value of the variance along such lines decreases quadratically with perpendicular distance from the origin. For seven-fold symmetry, it appears to vary approximately inversely with distance from the origin.