  Due to the weak birefringence of single mode fibers, solitons generated in
fiber lasers are indeed vector pulses and exhibit periodic parameter change
including polarization evolution even when there is a polarizer inside the
cavity. Period doubling eigenstates of solitons generated in a fiber laser
mode-locked by the nonlinear polarization rotation, i.e., period doubling of
polarization components of the soliton, are numerically explored in detail. We
found that, apart from the synchronous evolution between the two polarization
components, there exists asynchronous development depending on the detailed
operation conditions. In addition, period doubling of one polarization
component together with period-one of another polarization component can be
achieved. When the period tripling window is obtained, much complexed dynamics
on the two polarization components could be observed.
