Abstract
A recent experiment at the Indiana University Cyclotron Facility (IUCF) with electron cooling showed that rf phase modulation near 1:1 resonance leads to longitudinal beam splitting. Here we explain this by applying the method of averaging, a powerful tool from the study of dynamical systems, to the underlying equation of motion -- a pendulum equation with small damping and periodic forcing. The beam splitting is explained by showing that the associated Poincare map has two attracting fixed points, each with a well-defined basin of attraction. Our approach can be immediately applied to other accelerator physics problems governed by a similar equation.
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