  The motion of a relativistic test electron in a free electron laser can be
altered significantly by an ideal coaxial wiggler field and uniform axial guide
field. I have investigated the group I, group II and group III (the new group
that, it is found in coaxial wiggler) orbits and finally have found that those
groups become chaotic at sufficiently high beam density and at sufficiently
high amplitude of wiggler field. The threshold value of the wiggler amplitude
for the onset of chaos is estimated analytically and confirmed by computer
simulation for spatial case where self-field effects are negligibly small. It
is shown that the electron dynamic is nonintegrable. There is evidence for
chaos from numerical calculations of Poincare maps and nonzero Lyaponov
exponents using different approaches of Benetin's method which are described
and compared. Moreover, it is shown that the particle motion becomes chaotic on
a time scale comparable with the beam transit time through a few wiggler
periods.
