Structure and Breakdown of Invariant Tori in a 4-D Mapping Model of Accelerator Dynamics

We study sequences of periodic orbits and the associated phase space dynamics in a 4-D symplectic map of interest to the problem of beam stability in circular particle accelerators. The increasing period of these orbits is taken from a sequence of rational approximants to an incommensurate pair of irrational rotation numbers of an invariant torus. We find stable (elliptic–elliptic) periodic orbits of very high period and show that smooth rotational tori exist in their neighborhood, on which the motion is regular and bounded at large distances away from the origin. Perturbing these tori in parameter and/or initial condition space, we find either chains of smaller rotational tori or certain twisted tube-like tori of remarkable morphology. These tube-tori and tori chains have small scale chaotic motions in their surrounding vicinity and are formed about invariant curves of the 4-D map, which are either single loops or are composed of several disconnected loops, respectively. These smaller chaotic regions as well as the non-smoothness properties of large rotational tori under small perturbations, leading to eventual escape of orbits to infinity, are studied here by the computation of correlation dimension and Lyapunov exponents.