On the basis of the self-consistency equations, the chaotic behavior of electron orbits is studied numerically in a traveling-wave tube amplifier. The results show that motion orbits of electrons in phase space can become chaotic as the current increases, and the chaotic orbits are affected by the detuning. In temporal scale, appearance of chaotic motion of electrons is earlier than one of limit cycle and chaotic oscillation of the field. In comparison with the limit cycle oscillation of the field, the threshold current for the onset of chaotic orbits of electrons is low. In the soft nonlinear regime at which the field exhibits limit cycle oscillation, the chaotic region increases and gradually engulfs the whole phase space as the current increases. In the hard nonlinear regime at which the field exhibits chaotic behavior, the chaotic orbits of electrons occupy almost everywhere in the phase space. The pattern of chaotic motion of electrons is unchanged for a certain current; the characteristic of the limit cycle and chaotic oscillation of the field is certain, but their output power is uncertain in a certain current range.