An analysis of relativistic electron trajectories in a free-electron laser with a helical magnetic wiggler and an ion channel is presented. The wiggler field amplitude and the ion number density are taken to be uniform. Also included are the self-electric and self-magnetic fields of the electron beam, which is assumed to be of constant velocity and electron number density. The Hamiltonian, which is a constant of the motion, is first expressed in cartesian coordinates and momenta. A second constant of the motion is obtained by canonical transformation. The steady-state orbits, Poincaré maps, and Liapunov exponents are employed to investigate the chaotic motion in the presence of the ion channel. Numerical calculations reveal conditions under which chaotic and non-chaotic orbits exist.
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