The motion of a relativistic electron is analyzed in the field configuration consisting of a circular wiggler magnetic field, an axial magnetic field, and the equilibrium self-electric and self-magnetic fields produced by the non-neutral electron ring. By generating Poincare surface-of-section maps, it is shown that when the equilibrium self-fields is strong enough, the electron motions become chaotic. Although the realistic circular wiggler magnetic field destroys the inte-grability of the electron motion as the equilibrium self-fields do, the role the latter plays to make the motions become chaotic is stronger than the former does. In addition, the axial magnetic field can restrain the occurrence of the chaoticity.
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